Geometry Complete Unit 5 - High School Math Teachers...Unit 5 Pacing Chart...

Complete Unit 5

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HighSchoolMathTeachers.com©2020

Table of Contents

Unit 5 Pacing Chart -------------------------------------------------------------------------------------------- 1

Geometry Unit 5 Skills List ---------------------------------------------------------------------------------------- 5

Unit 5 Lesson Plans -------------------------------------------------------------------------------------------- 6

Day 66 Bellringer -------------------------------------------------------------------------------------------- 42

Day 66 Activity -------------------------------------------------------------------------------------------- 47

Day 66 Practice -------------------------------------------------------------------------------------------- 52

Day 66 Exit Slip -------------------------------------------------------------------------------------------- 56

Day 67 Bellringer -------------------------------------------------------------------------------------------- 58

Day 67 Activity -------------------------------------------------------------------------------------------- 60

Day 67 Practice -------------------------------------------------------------------------------------------- 63

Day 67 Exit Slip -------------------------------------------------------------------------------------------- 67

Day 68 Bellringer -------------------------------------------------------------------------------------------- 69

Day 68 Activity -------------------------------------------------------------------------------------------- 72

Day 68 Practice -------------------------------------------------------------------------------------------- 74

Day 68 Exit Slip -------------------------------------------------------------------------------------------- 78

Day 69 Bellringer -------------------------------------------------------------------------------------------- 80

Day 69 Activity -------------------------------------------------------------------------------------------- 82

Day 69 Practice -------------------------------------------------------------------------------------------- 85

Day 69 Exit Slip -------------------------------------------------------------------------------------------- 88

Week 14 Assessment -------------------------------------------------------------------------------------------- 90

Day 71 Bellringer -------------------------------------------------------------------------------------------- 97

Day 71 Activity -------------------------------------------------------------------------------------------- 100

Day 71 Practice -------------------------------------------------------------------------------------------- 104

Day 71 Exit Slip -------------------------------------------------------------------------------------------- 110

Day 72 Bellringer -------------------------------------------------------------------------------------------- 112

Day 72 Activity -------------------------------------------------------------------------------------------- 114

Day 72 Practice -------------------------------------------------------------------------------------------- 116

Day 72 Exit Slip -------------------------------------------------------------------------------------------- 119

Day 73 Bellringer -------------------------------------------------------------------------------------------- 121

Day 73 Activity -------------------------------------------------------------------------------------------- 124

Day 73 Practice -------------------------------------------------------------------------------------------- 127

Day 73 Exit Slip -------------------------------------------------------------------------------------------- 134

Day 74 Bellringer -------------------------------------------------------------------------------------------- 136

Day 74 Activity -------------------------------------------------------------------------------------------- 138

Day 74 Practice -------------------------------------------------------------------------------------------- 140

Day 74 Exit Slip -------------------------------------------------------------------------------------------- 144

Week 15 Assessment -------------------------------------------------------------------------------------------- 146

Day 76 Bellringer -------------------------------------------------------------------------------------------- 153

Day 76 Activity -------------------------------------------------------------------------------------------- 155

Day 76 Practice -------------------------------------------------------------------------------------------- 157

Day 76 Exit Slip -------------------------------------------------------------------------------------------- 161

Day 77 Bellringer -------------------------------------------------------------------------------------------- 163

Day 77 Activity -------------------------------------------------------------------------------------------- 165

Day 77 Practice -------------------------------------------------------------------------------------------- 168

Day 77 Exit Slip -------------------------------------------------------------------------------------------- 172

Day 78 Bellringer -------------------------------------------------------------------------------------------- 174

Day 78 Activity -------------------------------------------------------------------------------------------- 176

Day 78 Practice -------------------------------------------------------------------------------------------- 178

Day 78 Exit Slip -------------------------------------------------------------------------------------------- 183

Day 79 Bellringer -------------------------------------------------------------------------------------------- 185

Day 79 Activity -------------------------------------------------------------------------------------------- 187

Day 79 Practice -------------------------------------------------------------------------------------------- 189

Day 79 Exit Slip -------------------------------------------------------------------------------------------- 193

Week 16 Assessment -------------------------------------------------------------------------------------------- 195

Day 81 Bellringer -------------------------------------------------------------------------------------------- 202

Day 81 Activity -------------------------------------------------------------------------------------------- 204

Day 81 Practice -------------------------------------------------------------------------------------------- 206

Day 81 Exit Slip -------------------------------------------------------------------------------------------- 211

Day 82 Bellringer -------------------------------------------------------------------------------------------- 213

Day 82 Activity -------------------------------------------------------------------------------------------- 215

Day 82 Practice -------------------------------------------------------------------------------------------- 216

Day 82 Exit Slip -------------------------------------------------------------------------------------------- 223

Day 83 Bellringer -------------------------------------------------------------------------------------------- 225

Day 83 Activity -------------------------------------------------------------------------------------------- 227

Day 83 Practice -------------------------------------------------------------------------------------------- 229

Day 83 Exit Slip -------------------------------------------------------------------------------------------- 235

Day 84 Bellringer -------------------------------------------------------------------------------------------- 237

Day 84 Activity -------------------------------------------------------------------------------------------- 239

Day 84 Practice -------------------------------------------------------------------------------------------- 242

Day 84 Exit Slip -------------------------------------------------------------------------------------------- 246

Week 17 Assessment -------------------------------------------------------------------------------------------- 248

Unit 5 Test -------------------------------------------------------------------------------------------- 255

Unit 5 Pacing Chart

HighSchoolMathTeachers.com ©2020 Page 1

Unit Week Day CCSS Standards Objective I Can Statements

Unit 5 Similarity

Transformations

Week 14 – Midpoint Theorem

66

CCSS.MATH.CONTENT.HSG.CO.C.10 Prove theorems about triangles. Theorems include:

measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are

congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Show that when a line is bisected, the portions are

equal. Find the size of a line given a portion of the

bisected line. Find the size of each portion of the line

is bisected etc.

I can show that when a line is bisected, the portions are equal. I can find the size of a line given a

portion of the bisected line. I can find the size of each portion

of the line is bisected

Unit 5 Similarity

Transformations


67




Proving mid point theorem I can prove mid point theorem

Unit 5 Similarity

Transformations


68




Explain how midpoint theorem leads to dilation

I can explain how midpoint theorem gives rise of a dilation

Unit 5 Similarity

Transformations


69




Apply midpoint theorem to find length of other lines.

Use dilation

I can apply midpoint theorem to find length of other lines. Use

dilation

Unit 5 Pacing Chart


Unit 5 Similarity

Transformations


70 Assessment Assessment Assessment

Unit 5 Similarity

Transformations

Week 15 – Dilations

and Similarity

71

CCSS.MATH.CONTENT.HSG.SRT.A.1 Verify experimentally the properties of dilations

given by a center and a scale factor: CCSS.MATH.CONTENT.HSG.SRT.A.1.A

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line

passing through the center unchanged. CCSS.MATH.CONTENT.HSG.SRT.A.1.B

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Verify experimentally the properties of dilations given by a center and a

scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel

line, and leaves a line passing through the center

unchanged.

The dilation of a line segment is longer or

shorter in the ratio given by the scale factor.

I can verify experimentally the properties of dilations given by a

center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line

passing through the center unchanged.

The dilation of a line segment is

longer or shorter in the ratio given by the scale factor.

Unit 5 Similarity

Transformations


and Similarity

72

CCSS.MATH.CONTENT.HSG.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations

the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the

proportionality of all corresponding pairs of sides.

Given two figures, use the definition of similarity in

terms of similarity transformations to decide

if they are similar

Given two figures, I can use the definition of similarity in terms of

similarity transformations to decide if they are similar

Unit 5 Pacing Chart


Unit 5 Similarity

Transformations


and Similarity

73

CCSS.MATH.CONTENT.HSG.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations

the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the

proportionality of all corresponding pairs of sides.

Explain using similarity transformations the

meaning of similarity for triangles as the equality of all corresponding pairs of

angles and the proportionality of all

corresponding pairs of sides.

I can explain, using similarity transformations, the meaning of

similarity for triangles as the equality of all corresponding

pairs of angles and the proportionality of all

corresponding pairs of sides.

Unit 5 Similarity

Transformations


and Similarity

74

CCSS.MATH.CONTENT.HSG.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be

similar.

Use the properties of similarity transformations

to establish the AA criterion for two triangles

to be similar.

I can use the properties of similarity transformations to

establish the AA criterion for two triangles to be similar.

Unit 5 Similarity

Transformations


and Similarity


Unit 5 Similarity

Transformations

Week 16 – Prove

Theorems using

Similarity

76

CCSS.MATH.CONTENT.HSG.SRT.B.4 Prove theorems about triangles. Theorems include:

a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle

similarity.

Prove that a line parallel to one side of a triangle divides the other two

proportionally

I can prove that a line parallel to one side of a triangle divides the

other two proportionally

Unit 5 Pacing Chart


Unit 5 Similarity

Transformations

Week 16 – Prove

Theorems using

Similarity

77

CCSS.MATH.CONTENT.HSG.SRT.B.4 Prove theorems about triangles. Theorems include:

a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle

similarity.

Pythagorean Theorem proved using triangle

similarity.

I can prove Pythagorean Theorem using triangle similarity.

Unit 5 Similarity

Transformations

Week 16 – Prove

Theorems using

Similarity

78

CCSS.MATH.CONTENT.HSG.SRT.B.5 Use congruence and similarity criteria for triangles

to solve problems and to prove relationships in geometric figures.

Use congruence and similarity criteria for

triangles to solve problems

I can use congruence and similarity criteria for triangles to

solve problems

Unit 5 Similarity

Transformations

Week 16 – Prove

Theorems using

Similarity

79

CCSS.MATH.CONTENT.HSG.SRT.B.5 Use congruence and similarity criteria for triangles

to solve problems and to prove relationships in geometric figures.

Use congruence and similarity criteria for

triangles prove relationships in geometric

figures.

I can use congruence and similarity criteria for triangles

prove relationships in geometric figures.

Unit 5 Similarity

Transformations

Week 16 – Prove

Theorems using

Similarity


Geometry Unit 5 Skills List Name ____________________________________

HighSchoolMathTeachers.com©2020 Page 5

Geometry Unit 5 Skills List

Number Unit CCSS Skill

18 5 HSG.SRT.A.1 Verify the properties of dilations

19 5 HSG.SRT.A.2 Deciding whether two figures are

similar

20 5 HSG.SRT.A.3 Establish the AA criterion for two

triangles to be similar

21 5 HSG.SRT.B.4 Prove theorems about triangles

22 5 HSG.SRT.B.5 Use congruence and similarity criteria

for triangles to solve problems

23 5 HSG.SRT.B.5

Use congruence and similarity criteria

for triangles to prove relationships in

geometric figures

Unit 5 Lesson Plan Name ____________________________________


Unit: Unit 5 Similarity Transformations

Course: Geometry

Topic: Week 14 – Midpoint Theorem

Day: 66

Common Core State Standard: CCSS.MATH.CONTENT.HSG.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Mathematical Practice: CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use apprpriate tools strategically. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Objective: Show that when a line is bisected, the portions are equal. Find the size of a line given a portion of the bisected line. Find the size of each portion of the line is bisected etc

I can statement: I can show that when a line is bisected, the portions are equal. I can find the size of a line given a portion of the bisected line. I can find the size of each portion of the line is bisected

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. Students will construct a bisector to line and measure the resulting portions to see if they are equal

Materials: Bellringer 66 Day 66 Activities Day 66 Practice Day 66 Presentation Day 66 Exit Slip



3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.

Accommodations/Special Circumstances:

Technology:

Reflection:

Extra/Additional Resources:




Course: Geometry


Day: 67


Mathematical Practice: CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

Objective: Proving mid point theorem

I can statement: I can prove mid point theorem

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. Students will discover the triangle midpoint theorem by drawing a line joining the midpoints of any two sides of a triangle 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 68


Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Objective: Explain how midpoint theorem leads to dilation

I can statement: I can explain how midpoint theorem gives rise of a dilation

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will draw a triangle and a line passing through midpoints of two sides and compare the results with a dilation of the same triangle with a scale factor of 0.5 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 69


Mathematical Practice: CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use apprpriate tools strategically. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Objective: Apply midpoint theorem to find length of other lines. Use dilation

I can statement: I can apply midpoint theorem to find length of other lines. Use dilation

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 4. Students will create a framework from metal, plastic or wooden thin rods and verify the midpoint theorem. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 70

Common Core State Standard: Assessment

Mathematical Practice: Assessment

Objective: Assessment

I can statement: Assessment

Procedures: Assessment

Materials: Assessment


Technology:

Reflection:





Course: Geometry

Topic: Week 15 – Dilations and Similarity

Day: 71

Common Core State Standard: CCSS.MATH.CONTENT.HSG.SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: CCSS.MATH.CONTENT.HSG.SRT.A.1.A A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. CCSS.MATH.CONTENT.HSG.SRT.A.1.B The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quatitatively. CCSS.MATH.PRACTICE.MP5 Use apprpriate tools strategically. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

Objective: Verify experimentally the properties of dilations given by a center and a scale factor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

I can statement: I can verify experimentally the properties of dilations given by a center and a scale factor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Procedures:

Materials:



1. Students will complete the bellringer. 2. Students will work in groups of at least 4. They will verify that dilation takes a line segment that does not pass through the center of dilation to a segment parallel to the original segment and that dilation of a line segment is longer or shorter in the ratio given by the scale factor. They will verify by applying a dilation of scale factor 2, with center O, to three collinear points on a line 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.

Bellringer 71 Day 71 Activities Day 71 Practice Day 71 Presentation Day 71 Exit Slip


Technology:

Reflection:





Course: Geometry


Day: 72

Common Core State Standard: CCSS.MATH.CONTENT.HSG.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.


Objective: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar

I can statement: Given two figures, I can use the definition of similarity in terms of similarity transformations to decide if they are similar

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 6 (or even the whole class) to compare the similarity of their class and the rectangular block which the class is part of 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 73

Common Core State Standard: CCSS.MATH.CONTENT.HSG.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quatitatively. CCSS.MATH.PRACTICE.MP5 Use apprpriate tools strategically. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

Objective: Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

I can statement: I can explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will draw two similar triangles and identify their key properties.




3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.


Technology:

Reflection:





Course: Geometry


Day: 74

Common Core State Standard: CCSS.MATH.CONTENT.HSG.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.


Objective: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

I can statement: I can use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 4. students will construct similar triangles given ASA and prove that they are similar. Here AA criterion comes on due to AA’s in ASA postulate. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 75








Technology:

Reflection:





Course: Geometry

Topic: Week 16 – Prove Theorems using Similarity

Day: 76

Common Core State Standard: CCSS.MATH.CONTENT.HSG.SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.


Objective: Prove that a line parallel to one side of a triangle divides the other two proportionally

I can statement: I can prove that a line parallel to one side of a triangle divides the other two proportionally

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will draw a triangle of their choice and a line parallel to one of the sides then establish the proportionality of the parts of the sides divided by the parallel line. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 77



Objective: Pythagorean Theorem proved using triangle similarity.

I can statement: I can prove Pythagorean Theorem using triangle similarity.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will work in groups of four to verify the Pythagoras theorem from similar right triangles. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 78

Common Core State Standard: CCSS.MATH.CONTENT.HSG.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

Objective: Use congruence and similarity criteria for triangles to solve problems

I can statement: I can use congruence and similarity criteria for triangles to solve problems

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. Students will draw different triangles and identify the ones that are congruent. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:



Reflection:





Course: Geometry


Day: 79


Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

Objective: Use congruence and similarity criteria for triangles prove relationships in geometric figures.

I can statement: I can use congruence and similarity criteria for triangles prove relationships in geometric figures.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. Students will show that the diagonals of a rectangle bisect each other at the point of intersection. This is taken as a verification of the proof in the presentation. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 80








Technology:

Reflection:





Course: Geometry

Topic: Week 17 – Algebraic Proofs for Similarity

Day: 81



Objective: Show similarity using the idea that a line parallel to one side of a triangle divides the other two proportionally

I can statement: I can show Similarity using the idea that a line parallel to one side of a triangle divides the other two proportionally

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will draw a triangle, construct a line parallel to one side and use AA criterion to check whether the resulting triangles are similar 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.





Technology:

Reflection:





Course: Geometry


Day: 82



Objective: Similarity proved using Pythagorean Theorem

I can statement: I can prove similarity using Pythagorean Theorem

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 2.They will discover how the Pythagorean theorem yields similarity in right triangles 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:



Reflection:





Course: Geometry


Day: 83



Objective: Discuss reduction of figures

I can statement: I can discuss reduction of figures

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 4. They will two images under reduction and verify the scale factor under reduction 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:



Reflection:





Course: Geometry


Day: 84



Objective: Discuss enlargment of figures

I can statement: I can discuss the enlargment of figures

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. Students will enlarge a triangle on a coordinate plane with scale factor of -2 with the center of enlargement at the origin 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:



Reflection:


Day 66 Bellringer Name ____________________________________



Course: Geometry


Day: 85








Technology:

Reflection:




1. Construct a perpendicular bisector to the given lines.

a)

b)



c)

d)



2. Identify two sides that are congruent in the triangle below.

45°

45°

A B

C



Answer Key Day 66

1. a)

b)



c)

d)

2. AC and AB

Day 66 Activity Name ____________________________________


1. Draw a line of length 2𝑖𝑛 and label it AB.

2. Position the compass at end A and extend it to end B, then draw a circles as shown below.

A B



3. Position the compass at the end B and using the same compass width draw a circle as shown

A B



4. Join the intersections of the circles with a straight line as shown. Label the point where the two lines

meet as O.

A B O



5. Measure the length of AO. What do you get?

6. Measure the length of BO. What do you get?

7. Compare the two measurements in 5 and 6 above.



In this activity students will construct a bisector to line and measure the resulting portions to see if they

are equal. Students will work in groups of at least three and each group is required to have a compass, a

pencil, a ruler and a plane paper.

Answer Keys

Day 66:

1-4. No response

5. 2 𝑖𝑛

6. 2 𝑖𝑛

7. They are equal

Day 66 Practice Name ____________________________________


Use the diagram below to answer questions 1 and 2.

The point O is a bisector of the line QR. 𝑄𝑂 = 1.4𝑖𝑛

1. What is the length of OR?

2. Find the length of QR

Use the following information to answer questions 3 and 4.

A point S is a bisector to the line AB. The length of AB is 12in.

3. What is the length of SB?

4. What is the length of AS?

5. At what ratio does point S divide line AB?

Line RT divides another line UV in a ratio 1: 1. The two lines intersect at a point O.

The length of OV is 2.4𝑖𝑛. Use this information to answer questions 6 and 7.

6. What is the length of OU?

7. Find the length of the line UV.

O Q R



The line bisector of line CD intersects it at a point O. Line CD is 8𝑖𝑛 long. Use this information to answer

questions 8-10.

8. What is the length of CO?

9. What is the length of OD?

10. Write an equation relating OD and CD.

11. Write an equation relating OD and OC?

Use the following information to answer questions 12-14

A man wanted to erect a security light post exactly in the middle of his rectangular plot. The plot

measured 60ft by 80ft. In order to identify the middle of the plot he drew perpendicular bisectors to two

adjacent sides and erected it at the point of the intersection of the two bisectors.

12. What was the shortest distance from the post to the short side of the plot?

13. What was the shortest distance from the post to the long side of the plot?

14. What was the distance from one vertex of the plot to the post?

A point T divides line RK in the ratio 1:1. Line RK is 24in long.

Use this information to answere questions 15 to 17.

15. Find the length of RT.

16. Find the length of TK.



17. Find the RT:TK

18. Write the equation to relate RK and TK.

A line AB bisects line ST through at a point Q. Line ST 13in long.

Use this information to answer question 19 and 20.

19. What is the length of QT?

20. What is the length of QS?



Answer Keys

Day 66:

1. 1.4𝑖𝑛

2. 2.8𝑖𝑛

3. 6𝑖𝑛

4. 6𝑖𝑛

5. 1: 1

6. 2.4𝑖𝑛

7. 4.8𝑖𝑛

8. 4𝑖𝑛

9. 4𝑖𝑛

10. 𝐶𝐷 = 2𝑂𝐷

11. 𝑂𝐶 = 𝐶𝐷

12. 40 ft

13. 30 ft

14. 50ft

15. 12𝑖𝑛

16. 12𝑖𝑛

17. 2: 1

18. RK=2TK

19. 6.5𝑖𝑛

20. 6.5𝑖𝑛

Day 66 Exit Slip Name ____________________________________


1. Line PQ of length 10in is bisected by a point O. What is the sum of the lengths of the two portions?



Answer Keys

Day 66:

1. 10𝑖𝑛



Consider ∆KLM (not drawn to scale) shown below where NP = 2.3 in, LN = 3.1 in, NQ = 4.2 in and

KQNP is a parallelogram. Use it to answer the following questions.

(a) Given that KQNP is a parallelogram, find the length KP.

(b) Hence find the length KM.

(c) Find the length KQ on parallelogram KQNP

(d) Hence find the length KL.

(e) Compare the length of KL to that of NP. What do you notice?

K L

M

N P

Q

2.3 in

3.1 in 4.2 in



Answer keys Day 67:

(a) KP = 4.2 in.

(b) KM = 8.4 in.

(c) KQ = 2.3 in.

(d) KL = 4.6 in

(e) The length of NP is half that of KL



1. On the plain paper, draw a triangle of suitable size and label it ΔKLM just like the triangle shown

below.

2. Measure the length of KM̅̅̅̅̅ and a hence carefully locate point N, the midpoint of KM̅̅̅̅̅ as shown below.

3. Similarly, measure the length of ML̅̅ ̅̅ and a hence carefully locate point P, the midpoint of ML̅̅ ̅̅ as

shown below.

M

K L

M

K L

N

M

K L

N P



4. Join point N to point P using a ruler as shown below.

5. Measure the lengths of NP̅̅ ̅̅ and KL̅̅̅̅ . What do you notice after comparing these lengths?

6. Now, join point P to point K using a broken line as shown below.

7. Measure ∠NPK and ∠LKP then compare their measures. What main conclusion can be drawn about

the relationship between NP̅̅ ̅̅ and KL̅̅̅̅ ?

M

K L

N P

M

K L

N P



In this activity students will discover the triangle midpoint theorem by drawing a line joining the

midpoints of any two sides of a triangle. Students can work in groups of three or four. Each group should

have a plain paper, a ruler and a protractor.

Answer keys Day 67:

1. No response

2. Ensure that KN ≅ MN

3. Ensure that MP ≅ LP

4. No response

5. The length of NP̅̅ ̅̅ is half that KL̅̅̅̅

6. No response

7. NP̅̅ ̅̅ ∥ KL̅̅̅̅



Use the figure below to answer questions 1 - 8.

In the figure, S is the midpoint of PQ, ST is parallel to QR, SU is parallel to PR, ∠𝑃𝑆𝑇 = 58° and ∠𝑃𝑅𝑄 =

58°.

1. Find the measure of ∠𝑆𝑄𝑈

2. Find the measure of ∠𝑄𝑈𝑆

3. Find the measure of ∠𝑃𝑇𝑆

4. Find the measure of ∠𝑇𝑃𝑆

5. Find the measure of ∠𝑄𝑆𝑈

6. Considering the angle measures you have found in questions. State whether Δ𝑃𝑆𝑇 is congruent to

Δ𝑆𝑄𝑈 or not.

P

Q R

S T

U



7. Given that TS ≅ RU, compare the length TS to the length RQ.

8. Considering your answer in question 7 above, is U is the midpoint of QR?

Use the figure below to answer questions 9-12.

In ΔKLM below, N, Q and P are the midpoints of KL, LM and KM respectively and 𝐿𝑁 ∥ 𝑄𝑃. Show that

LQPN is a parallelogram by completing the table below.

Statement Reasons

𝐿𝑁 ∥ 𝑄𝑃 9.

𝑁𝑃 ∥ 𝐿𝑄 10.

11. QP ≅1

2LK but LN ≅

1

2LK

LQ ≅ NP

12.

K

L M

N P

Q



Use the figure below to answer questions 13-20. E and D are the midpoints of AB and AC respectively,

𝐸𝐷 ≅ 𝐷𝐹 and 𝐶𝐹 ∥ 𝐵𝐸

Given that E and D are the midpoints of AB and AC respectively, give two relationships between ED and

BC in the table below?

13.

14.

Fill in the table below.

Measure of ∠𝑪𝑫𝑭 Reason

15. 16.

Find the measures of the following pairs of angles:

17.∠𝐶𝐵𝐸, hence ∠𝐵𝐸𝐷

18.∠𝐵𝐶𝐹, hence ∠𝐶𝐹𝐷

19. Compare the pairs of angles in question 17 and 18. What do you notice?

20. What type of quadrilateral is most likely to be BCFE according to the angles you have found

questions 17 and 18 above?

A

B C

D E F 59°

61°



Answer keys Day 67:

1. 58°

2. 58°

3. 58°

4. 64°

5. 64°

6. They are congruent

7. TS ≅1

2RQ/ RQ ≅ 2TS

8. Yes

13. ED ≅

1

2BC / BC ≅ 2ED

14. 𝐸𝐷 ∥ 𝐵𝐶

Measure of ∠𝑪𝑫𝑭 Reason

15. 59° 16. Vertical angles / Opposite angles

17. ∠𝐶𝐵𝐸 = 60°, ∠𝐵𝐸𝐷 = 120°

18. ∠𝐵𝐶𝐹 = 120°, ∠𝐶𝐹𝐷 = 60°

19. The angles in each pair are congruent

20. A parallelogram

Statement Reasons

𝐿𝑁 ∥ 𝑄𝑃 9. Triangle midpoint theorem

𝑁𝑃 ∥ 𝐿𝑄 10. Triangle midpoint theorem

11. 𝑳𝑵 ≅ 𝑸𝑷 QP ≅1

2LK but LN ≅

1

2LK

LQ ≅ NP 12. 𝐍𝐏 ≅𝟏

𝟐𝐋𝐌 but 𝐋𝐐 ≅

𝟏

𝟐𝐋𝐌



In the figure below TYUS is a parallelogram in ΔXYZ. YU̅̅̅̅ is produced to point Z such that UY ≅ UZ and

XS̅̅ ̅ is also produced to point Z. Show that XS ≅ ZS by completing the table below the triangle.

Statement Reason

TY ∥ SU

U is the midpoint of YZ if follows that S is also the midpoint of XZ

Hence XS ≅ ZS

X

Y Z

T

U

S



Answer keys Day 67:

Statements Reasons

TY ∥ SU Opposite sides of a parallelogram are parallel

U is the midpoint of YZ if follows that S is also

the midpoint of XZ

Triangle midpoint theorem

Hence XS ≅ ZS



1. Use the figure below to answer the questions that follow. Points S and T are the midpoints of sides AC

and BC respectively. ST= 2.5𝑖𝑛

a) What is the size of ∆𝐴𝐵𝐶?

b) What is the length of side AB?

127°

A B

C

S T



2. Use the figure below to answer the questions that follow. ∆𝐴𝐵𝐶 is dilated to form ∆𝐶𝐷𝐸.

a) Find the scale factor of dilation.

b) Where is the center of dilation?

c) If side AC is 2.8𝑖𝑛 long, what is length of CD?

A

B

C

D

E

1.5𝑖𝑛 3𝑖𝑛



Answer Key

Day 68:

1. a) 53°

b) 5𝑖𝑛

2. a) 1

2

b) Point C

c) 1.4𝑖𝑛



1. Plot a graph with a scale of 1square representing 1 unit as shown below

2. Draw a triangle with its vertices at points (0,0) 𝐵(5,1), 𝐶(1,4).

3. Identify the midpoints of sides AB and AC and label them as D and E respectively.

4. Join D and E with a straight line.

What are the vertices of ∆𝐴𝐷𝐸?

5. Dilate ∆𝐴𝐵𝐶 with a scale factor of 1

2.

Does the triangle resulting from the dilation coincide with ∆𝐴𝐷𝐸?

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4



In this activity students will draw a triangle and a line passing through midpoints of two sides and

compare the results with a dilation of the same triangle with a scale factor of 0.5. Students will work in

groups of at least three and each group is required to have a graph paper, a pencil and a ruler.

Answer Keys

Day 68:

1-3. No response

4. 𝐴(0,0), 𝐷(2.5,0.5), 𝐸(0.5,2)

5. Yes



Use the diagram below to answer questions 1-5.

AB is parallel to ST. 𝑆𝑈 = 8𝑖𝑛, ST= 6𝑖𝑛, and AB= 3𝑖𝑛. (the figure is not drawn to scale)

1. Which transformation will map ∆𝑆𝑇𝑈 onto ∆𝐴𝐵𝑈?

2. What is the length of AU?

3. What is the length of AS?

4. What is the length of BU?

5. What is the length of BT?


𝐴𝐶 ∥ 𝑅𝑇 and RT is twice the length of RT.

𝑆 T

U

A B

6 𝑖𝑛

3𝑖𝑛

5𝑖𝑛

𝑅

A

B C

T



6. Which transformation will map ∆𝐴𝐵𝐶 onto ∆𝑇𝐵𝑅?

7. What is the length of CR?

8. What is the length of BR?

9. What is the length of BT?

10. What is the length of AT?

Use the figure below to answer questions 11 to 15.

RQ is parallel to MN and NO=MO=10 in.

11. Which geometric transformation will map ∆𝑀𝑁𝑂 onto ∆𝑄𝑅𝑂?

12. What is the length of RO?

13. What is the length of MR?

14. What is the length of OQ?

15. What is the length of NQ?

12𝑖𝑛

Q R

O

N M

6𝑖𝑛




𝐾𝐽 =1

2𝐵𝐶.

16. Which geometric transformation will map ∆𝐴𝐵𝐶 onto ∆𝐴𝐽𝐾?

17. Write an equation that relates AK and CK.

18. Write an equation that relates AK and AC.

19. Write an equation that relates AJ and BJ.

20. Write an equation that relates BJ and AB.

A B

C

K

J



Answer Keys

Day 68:

1. A dilation about point U with a scale factor of 1

2

2. 4𝑖𝑛

3. 4𝑖𝑛

4. 5𝑖𝑛

5. 5𝑖𝑛

6. A dilation about point B with a scale factor of 1

2

7. 1.5𝑖𝑛

8. 1.5𝑖𝑛

9. 2.5𝑖𝑛

10. 2.5𝑖𝑛

11. A dilation about point B with a scale factor of 1

2

12. 5𝑖𝑛

13. 5𝑖𝑛

14. 5𝑖𝑛

15. 5𝑖𝑛

16. A dilation about point C with a scale factor of 1

2

17. 𝐴𝐾 = 𝐶𝐾

18. 𝐶𝐾 =1

2𝐴𝐶

19. 𝐴𝐽 = 𝐵𝐽

20. 𝐵𝐽 =1

2𝐴𝐵



1. Points D and E are midpoints of AB and BC respectively. ∆𝐵𝐷𝐸 is a dilation of ∆𝐴𝐵𝐶. What is the scale

factor of dilation?

A B

C

D

E



Answer Keys

Day 68:

1. 1

2



Use the following diagram to answer the questions

Line AJ and FH are parallel. Line NG = 3.5 in and NG = 2MG. Angle GKH =116° and Angle AMS = 42°.

1. Find the length of MN

2. Find the size of angle GTJ.

3. Find the size of angle GTJ.

4. Find the size of angle GNH.

5. Find the length of MN

A

F

G

H

J T

M

N

K

S



Answer Keys

Day 69:

1. 3.5 in

2. 116°

3. 42°

4. 42°

5. 7 in



1. Using 3 rods make a triangle of suitable size by connecting the rods with strings.

2. Label the vertices of the triangle as ABC and measure their length.

3. What is the dimensions of the triangle?

4. Pick any two sides and identify their midpoint using a ruler

5. Tie the last rod to form a line from one midpoint to the other.

6. Measure the distance between these midpoints



7. Compare the distance in 6 above with the third side of the triangle ( the side whose midpoint was not

identified in 4 above).

8. Find the shortest distance between the third side in 7 above and the line connecting the midpoints.

9. Is the shortest distance constant throughout the two rods?

10. What do you conclude based on 9 above.



In this activity, students will work in groups of at least 4. They will create a framework from metal,

plastic or wooden thin rods and verify the midpoint theorem. Each group will require a 4 five metal,

wooden or plastic rods or length 5 – 7 inches each, strings and a ruler.

Answer Keys

Day 69:

1. No response

2. The vertices are labelled

3. Answers varies

4 -5.No response

6. Answer varies

7. The third side must be approximately twice line connecting the midpoint

8. No response

9. Yes

10. The Third side and the line connecting the midpoints are parallel



Use the following information to answer questions 1 – 8

In the figure below, N divides LQ in the ratio of 1:1 while PM = 𝑃𝑀 = 𝑀𝑄. A point T is on line PL such

that 𝑃𝐿 = 2𝑇𝐿. 𝑇𝐿 = 3 𝑖𝑛, 𝐿𝑁 = 5.5 𝑖𝑛 and 𝑃𝑄 = 9 𝑖𝑛. Angle 𝑃𝑇𝑀 = 27°.

1. Which kind of lines are PL and MN?

2. Find the relationship between the two lines above.

3. What is the measurement of MN.

4. What is the measurement of PL.

5. What is the measurement of MQ

6. What is the measurement of LQ.

7. What kind of lines are MN and TL if any.

8. Is there any linear relationship between the two lines in 7 above?

9. What kind of figure is TMNL?

P

Q

L

M

N



10. Explain your results above.

11. Find the size of angle LNM.

12. Find the side of angle MNQ

13. Find the size of angle PLN.

Use the following diagram to answer the following questions. 14 – 20.

In the diagram below, HK and D are midpoints of AG, GC and CE respectively.

14. Compare the length of HK and AC.

15. Compare the length of HK and AB.

16. Compare the length of KD and GE.

17. Provide the reason for your answer in 16 above.

18. If HKD is a straight line and K divides HD twice, what type of lines are HD and GE?

19. What is the linear relationship between GE and AC.

20. What kind of figure is KDEF.

A B C

D

E F G

H

K



Answer keys Day 69:

1. Parallel lines

2. 𝑃𝐿 = 2𝑀𝑁 3. 3 in

4. 6 in

5. 4.5 in

6. 11 in

7. They are parallel

8. 𝑀𝑁 = 𝑇𝐿 9. Parallelogram

10. MN and TL are parallel while LN and TM are parallel too, Opposite sides are parallel

11. 153°

12. 27°

13. 27°

14. 𝐻𝐾 =1

2𝐴𝐶

15. 𝐻𝐾 = 𝐴𝐵

16. 2𝐾𝐷 = 𝐺𝐸 17. Due to mid-point theorem since K and D are midpoints of CE and CG respectively

18. 𝐻𝐷 = 𝐺𝐸

19. 𝐺𝐸 = 𝐴𝐶 20. Parallelogram



In the figure below, R and E are the midpoints of YP and YH respectively. Find the size of angle REH.

P H

Y

R E

43°



Answer Keys

Day 69

137°

90

High School Math Teachers

Geometry

Weekly Assessment Package

Week 14

©2020HighSchoolMathTeachers

91

Week 14

Weekly Assessment

92

Week #14 1. A line bisector of line AB is drawn such that

it is 3.1 𝑖𝑛 from point B.

a) Find the length of AB. b) If it intersects AB at O, what is the length of OA?

2. Use the diagram below to answer the questions that

follow.

a) Find the length of AB.

b) Find the size of ∠𝐴𝑂𝑁.

3. In the figure below, points A and B are the

midpoints of OT and OS respectively.

𝑆𝑇 = 3.2 𝑖𝑛. a) Find the scale factor of dilation. b) Find the length of AB.

4. Use the diagram below to answer the questions that

follow.

a) Which postulate shows that the two triangles are congruent? b) Find the value of 𝑥.

A B

C

O

M N 1.2 𝑖𝑛

36°

T

S

O A

B

(2𝑥 − 3) 𝑖𝑛

(6𝑥) 𝑖𝑛

93

5. Use the diagram below to answer the questions that follow.

a) Which postulate shows that the two triangles are congruent? b) Find the value of value of 𝑥. 6. Use the diagram below to answer the questions that follow.

a) Which rigid motion will map the two triangles onto each other. b) Are the two triangles congruent to each other.

(20𝑥)°

(25𝑥 − 25)°

94

Week 14 - Keys

Weekly Assessment

95

Week #14 KEY 1. A line bisector of line AB is drawn such that it is 3.4 𝑖𝑛 from point B. a) Find the length of AB. 6.8 𝑖𝑛 b) If it intersects AB at O, what is the length of OA? 3.4 𝑖𝑛


a) Find the length of AB.

2.4 𝑖𝑛

b) Find the size of ∠𝐴𝑂𝑁. 144°

3. In the figure below, points A and B are the midpoints of OT and OS respectively.

𝑆𝑇 = 3.2 𝑖𝑛. a) Find the scale factor of dilation.

1

2

b) Find the length of AB. 1.6 𝑖𝑛

4. Use the diagram below to answer the questions that follow. a) Which postulate shows that the two triangles are congruent? ASA postulate b) Find the value of 𝑥. 2

A B

C

O

M N 1.2 𝑖𝑛

36°

T

S

O A

B

(2𝑥 + 8) 𝑖𝑛

(6𝑥) 𝑖𝑛

96


a) Which postulate shows that the two triangles are congruent? SAS postulate b) Find the value of value of 𝑥. 5 6. Use the diagram below to answer the questions that follow. a) Which rigid motion will map the two triangles onto each other. Reflection b) Are the two triangles congruent to each other. Yes

(20𝑥)°

(25𝑥 − 25)°



A dilation about the origin is shown on the graph below. ΔKLM has been dilated to form its

image, ΔK′L′M′.

1. Given that one centimeter on the graph represents one unit, find the lengths (in units) of the

following sides on the triangles.

(a) KL̅̅̅̅

(b) K′L′̅̅ ̅̅ ̅

(c) LM̅̅ ̅̅

𝑥

𝑦

0 1 2 3 4 5 6 7 8 9 10 11 12 13

9

8

7

6

5

4

3

K

L M

K′

L′ M′



(d) L′M′̅̅ ̅̅ ̅̅

2. Calculate the ratios of the following corresponding sides

(a) K′L′̅̅ ̅̅ ̅̅

KL̅̅ ̅̅

(b) L′M′̅̅ ̅̅ ̅̅ ̅

LM̅̅ ̅̅̅

3. Hence find the scale factor of the dilation.



Answer keys

Day 71:

1. (a) 3 units

(b) 6 units

(c) 3 units

(d) 6 units

2. (a) 2

(b) 2

3. The scale factor is 2



1. Apply a dilation of scale factor 2 with O as the center on the three points P, Q and R to locate points

P′, Q′and R′ the images of P, Q and R respectively. Follow the steps below:

(a) On the diagram provided, use a ruler to draw broken lines from O through P, Q and R as shown

below.

(b) Use a ruler to measure the lengths OP, OQ and OR.

(c) Locate P′ on the broken line through P such that the distance from O to P′ is twice (scale factor 2) the

distance from O to P.

O

P Q R

O

P Q R



(d) Now, locate Q′ and R′ on the broken lines through Q and R respectively using a procedure similar to

that you have used to locate P′ in (c) above. The relative positions of the three points are shown below.

(e) Draw a line passing through the points P′, Q′and R′ points as shown below.

2. Measure the length of segments PR and P′R′. What is the relationship between the length PR and the

length P′R′?

O

P Q R

P′ Q′ R′

O

P Q R

P′ Q′ R′



3. Similarly, measure the length of segments PQ and P′Q′. What is the relationship between the length

PQ and the length P′Q′?

4. Use a protractor to measure ∠P′Q′Q and ∠Q′QR and compare the measures. What does this suggest

about the relationship between segments PR and P′R′?



In this activity students will verify that dilation takes a line segment that does not pass through the

center of dilation to a segment parallel to the original segment and that dilation of a line segment is

longer or shorter in the ratio given by the scale factor. They will verify by applying a dilation of scale

factor 2, with center O, to three collinear points on a line. Students can work in groups of at least 4. The

groups should be equipped with rulers, pencils, protractors and a copy of the diagram below.

Answer keys Day 71:

1. No response

2. 𝑃′𝑅′ ≅ 2𝑃𝑅

3. 𝑃′𝑄′ ≅ 2𝑃𝑄

4. 𝑃𝑅 ∥ 𝑃′𝑅′



Use the grid below to answer question 1-8.

Points P, O and Q lie on line 𝑙. Locate the positions of the image of the following points on the grid after

a dilation of scale factor 1.5 with point O as your center of dilation

1. P

2. O

3. Q

4. Describe the positions of points 𝑃′, 𝑂′ and 𝑄′ with reference to the positions of points P, O and Q.

5. What can you say about the positions of segments PQ̅̅̅̅ and P′Q′̅̅ ̅̅ ̅?

P O Q 𝒍



6. How many times is P′Q′̅̅ ̅̅ ̅̅ longer than PQ̅̅̅̅ ?

7. According to question 6 above, state whether the number of times P′Q′̅̅ ̅̅ ̅̅ is longer than PQ̅̅̅̅ , is equal to

the scale factor of dilation or not?

8. What would be the difference in P′Q′̅̅ ̅̅ ̅̅ , the image of PQ̅̅̅̅ after a dilation whose center is not on line 𝑙?

Use the grid below to answer questions 9-15.

Points X and Y are collinear. Draw X′Y′̅̅ ̅̅ ̅, the image of XY̅̅̅̅ by locating the positions of points of the

following points after a dilation with scale factor 3 about the point O.

9. 𝑋′

10. 𝑌′

X

O

Y



Given that one side of a square on the grid represents one unit, find the length of:

11. XY̅̅̅̅

12. X′Y′̅̅ ̅̅ ̅

13. Find the number of times X′Y′̅̅ ̅̅ ̅ is longer than XY̅̅̅̅ .

14. Compare the number of times you have found in question 13 above to the scale factor of the

dilation. What do you notice?

15. From the grid above, briefly give two relationships between XY̅̅̅̅ and X′Y′̅̅ ̅̅ ̅?



Use the graph below to answer questions 16-19.

On the graph below ΔJKL has been mapped onto ΔJ′K′L′ after a dilation whose center is the point (4,2).

One centimeter on the graph represents one unit.

Regard the side KL and its corresponding side K′L′ as line segments. Find the length (in units) of:

16. KL

17. K′L′

18. Hence find the scale factor of the dilation above.

19. What can you say about the position of KL and K′L′ with reference to the center of dilation?

20. If a line segment PQ, 6 inches long, is dilated to form its image P′Q′ with a scale factor of 0.6, what

will be the length of P′Q′?

𝑥

𝑦

0 1 2 3 4 5 6 7 8 9 10 11 12 13

9

8

7

6

5

4

3

J

K L

J′

K′ L′



Answer keys

Day 71:

4. Points P′, O′ and Q′ all lie on line 𝑙, just like points P, O and Q

5. Both P′Q′̅̅ ̅̅ ̅̅ and PQ̅̅̅̅ lie on line 𝑙

6. 1.5 times longer

7. It is equal to the scale factor of the dilation

8. P′Q′̅̅ ̅̅ ̅̅ would be parallel to PQ̅̅̅̅ but not on line 𝑙

P O Q 𝒍 P′ Q′ O′



11. 3 units

12. 9 units

13. 3 times longer

14. The number of times is equal to the scale factor

15. X′Y′ = 3XY and X′Y′ ∥ XY

16. 6 units

17. 3 units

18. Scale factor = 0.5

19. KL, K′L′ and the center of dilation all lie on the same line.

20. 3.6 inches

X

O

Y

X′ Y′



On the grid provided, draw X′Y′̅̅ ̅̅ ̅, the image of XY̅̅̅̅ after a dilation with scale factor 2 about the point Z.

Briefly give two major visible relationships between X′Y′̅̅ ̅̅ ̅and XY̅̅̅̅

X Y

Z



Answer keys

Day 71:

X′Y′̅̅ ̅̅ ̅ ∥ XY̅̅̅̅ and X′Y′̅̅ ̅̅ ̅ = 2XY̅̅̅̅

X Y

Z

X′ Y′



I. Find the image point or length of image under the following dilations.

1. (3,4) dilated by a scale factor of 2.5.

2. (0,1.5) dilated by a scale factor of 2.0

3. (1.5, 3.2) dilated by a scale factor of -1.2 with center of dilation at the origin

4. A line of length 3.2 in dilated by a scale factor of 0.4

5. A line of length 1.8 in dilated by a scale factor of 3.4

II. Which side on ZPR corresponds to AB.

A

B

C

R

T

K



Answer Keys

Day 72:

I. 1. (7.5,10)

2. (0,3.0)

3. (−1.8, −3.84)

4. 1.28

5. 6.12

II. KR



1. Measure the length and width of the classroom from outside.

2. Measure the length and the width of the rectangular block which the class is part of.

3. Find the fraction 𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑙𝑜𝑐𝑘

𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠

4. Find the fraction 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑙𝑜𝑐𝑘

𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠

5. Are the two fractions in 3 and 4 above equal?

6. Given that the class was also a rectangle, what is the assumption made based on the relation of the

angles of the two; the class and the block?

7. Which conclusion about similarity can be made from the two?



In this activity, students will work in groups of at least 6 (or even the whole class) to compare the

similarity of their class and the rectangular block which the class is part of. Each group will require a field

ruler, a book to take records and a pen.

Answer Keys

Day 72:

1-5.Different responses

6. Corresponding angles are equal

7. Only two possibilities: similar or not similar




1. Identify corresponding sides in the figures above.

2. Identify a rigid motion(s) present between the two figures above.

3. Calculate the scale factor relating the two triangles given that TQR is the image of WEX.

4. Which kind of dilation relate the image and its pre-image above?

5. Are the two related by similarity transformation? Explain.

W

X

E

9

4 in

6

i

T

R Q

4.5

2 in

3

in



Use the following figure to answer questions 6 – 20.

We wish to proof that the there is a similarity transformation between the two images below if DR and

GH are parallel lines and corresponding sides of triangle YGH and YDR are proportional.

Statement Reason

DR and GH are parallel 6.

GD and RH are transversal line 7.

∠𝑌𝐺𝐻 = ∠𝑌𝐷𝑅 8.

9. Corresponding angles

∠𝐺𝑌𝐻 = ∠𝐷𝑌𝑅 10.

The two triangles are have the same shape 11.

One triangle is translated to another by zero units.

12.

Corresponding sides are proportional 13.

One triangle is an image of another due to dilation

14.

A similarity transformation exists between the two triangles

15.

16.𝐺𝐻

𝐷𝑅=

𝑌𝑅

Corresponding sides are proportional

17. RH = ____________ Upon solving the problem in 16 above.

18.𝐺𝐻

𝐷𝑅=

𝑌𝐷


19. YD = ____________ Upon solving the problem in 19 above.

20. GY =____________ GD is the sum of YD and GD

G H

Y

D R

16 in

6 in

4 in

3 in



Answer keys Day 72:

1. XE and RQ

EW and QT WX and TR

2. Translation

3. 1

2

4. Reduction

5. The size of the image is smaller than that of the pre-image

Statement Reason

DR and GH are parallel 6. Given

GD and RH are transversal line 7. They intersect parallel lines

∠𝑌𝐺𝐻 = ∠𝑌𝐷𝑅 8. Corresponding angles

9. ∠𝑌𝐻𝐺 = ∠𝑌𝑅𝐷

Corresponding angles

∠𝐺𝑌𝐻 = ∠𝐷𝑌𝑅 10. Common to both triangles

The two triangles are have the same shape 11. Corresponding angles are equal

One triangle is translated to another by zero units.

12. They have the same orientation and no evidence of movement is shown

Corresponding sides are proportional 13. Given

One triangle is an image of another due to dilation

14. Corresponding angles are equal and corresponding sides are proportional

A similarity transformation exists between the two triangles

15. There is a rigid motion and a dilation between the two figures

16.𝐺𝐻

𝐷𝑅=

𝑌𝐻

𝑌𝑅


17. RH = ______20

3𝑖𝑛______ Upon solving the problem in 16 above.

18.𝐺𝐻

𝐷𝑅=

𝑌𝐺

𝑌𝐷


19. YD = _____9

5 𝑖𝑛 𝑜𝑟 1.8 𝑖𝑛_______ Upon solving the problem in 19 above.

20. GY =_____4.8 in_______ GD is the sum of YD and GD



Given two figures where one is an image of another under rotation, are they similar? Explain



Answer Keys

Day 72

Yes

At least one rigid motion (rotation) exists

A dilation of scale factor 1 exists since the images and the object are of the same size



The figure below shows ΔRST and its image, ΔR′S′T′ after a dilation of scale factor 1

2 about the point O.

∠RTS = 36.9° (All measurements are given in inches and the diagram is not drawn to scale)

(a) Find the length RS̅̅̅̅ on ΔRST

(b) Find the length S′T′̅̅ ̅̅ ̅ on ΔR′S′T′

R

S T

R′

S′ T′

O 6

16

10

36.9°



(c) Find the length RT̅̅̅̅ on ΔRST

(d) Evaluate the following ratios of corresponding sides: R′S′

RS,

S′T′

ST and

R′T′

RT.

(e) Basing on (d) above, what conclusion can be drawn about ratios of corresponding sides after a plane

figure is dilated?

(f) Using the properties of a dilation, find the measure of angle ∠R′T′S′ on ΔR′S′T′.



Answer keys Day 73:

(a) 12 inches

(b) 8 inches

(c) 20 inches

(d) R′S′

RS=

S′T′

ST=

R′T′

RT=

1

2

(e) All ratios of corresponding sides are equal

(f) ∠R′T′S′ = 36.9°



1. Draw any triangle of suitable size using a ruler on the blank paper provided. Label it as shown below.

2. Mark any point about the midpoint of AB̅̅ ̅̅ and label it D as shown below.

3. Using a compass construct a line parallel to BC̅̅̅̅ through point D to intersect AC̅̅̅̅ at point E as shown

below.

4. Using one of the tracing papers provided and with the help of a ruler, carefully trace out ΔADE.

A

B C

A

B C

D

A

B C

D E



5. Use the other tracing paper to carefully trace out ΔABC.

6. Compare the shape of ΔADE to that of ΔABC. What do you notice?

7. Identify one angle common between the two triangles and find its measure.

8. Measure ∠𝐷 on ΔADE and compare it to ∠𝐵 on ΔABC. What do you notice?

9. Measure ∠𝐸 on ΔADE and compare it to ∠𝐶 on ΔABC. What do you notice?

10. What conclusion can be drawn from the corresponding angles you have measured in relation to the

two triangles?

11. Measure the length of AB̅̅ ̅̅ and AD̅̅ ̅̅ and hence find the ratio AB̅̅ ̅̅

AD̅̅̅̅̅

12. Measure the length of AC̅̅̅̅ and AE̅̅̅̅ and hence find the ratio AC̅̅ ̅̅

AE̅̅ ̅̅

13. Measure the length of BC̅̅̅̅ and DE̅̅ ̅̅ and hence find the ratio BC̅̅ ̅̅

DE̅̅ ̅̅

14. Compare the ratios AB̅̅ ̅̅

AD̅̅̅̅̅,

AC̅̅ ̅̅

AE̅̅ ̅̅ and

BC̅̅ ̅̅

DE̅̅ ̅̅. What do you discover about these ratios?



In this activity students will work in small groups of three to four to draw two similar triangles and

identify their key properties. Each group will require a ruler, a pencil, a protractor, a compass, two

tracing papers and a blank paper to perform the construction. It is presumed that the students have

dealt with construction of parallel lines in earlier lessons.

Answer keys

Day 73:

1. No response

2. No response

3. Accuracy should be emphasized

4. No response

5. No response

6. They have the same shape

7. ∠𝐴. The angle measure will vary

8. ∠𝐷 ≅ ∠𝐵. The angle measures will vary

9. ∠𝐸 ≅ ∠𝐶. The angle measures will vary

10. Corresponding angles are congruent

11. The lengths will vary



14. The ratios are equal



1. An elephant is 12 feet tall and its shadow is 6 feet long at exactly 10.00 am. Ken’s shadow is 3 feet at

the same time. Find the Ken’s height using the concept of similar triangles.

2. I wish to reduce without changing the shape of my triangular table mat which has sides of length 8, 8

and 10 inches so that it fits on my new table. If the longest side should be 5 inches, how long should the

other sides be?

3. When a building casts a shadow 60 feet long, a man 5 feet tall casts a shadow of length 6 feet. How

tall is the building?

4. A carpenter wants to reduce by scaling down a triangular ply board measuring sides measuring 8, 10

and 12 inches so that it fits on a triangular frame whose shortest side is 4 inches. How long should the

other two sides be?

5. An electrician leans a 10 foot-ladder against a wall in such a way that the ladder touches the wall at a

height of 5 feet above the ground. If he decides to use a 15 foot-ladder and leans it against the wall such

that both ladders form the same angle with the ground, at what height above the ground will this ladder

touch?



For the following pairs of similar triangles in questions 6-10, find the missing side represented by the

letter. The triangles are not drawn to scale and all measurements are inches.

6.

7.

8.

12

8 𝑥

20

2

10

16

3.2

𝑥

4

10

12

𝑥

12

5



9.

10.

4

𝑥

16

14

30

30

8

𝑥

18

2



Study and use the figure below to answer questions 11-15. ∆ABC~ΔDEF

Identify an angle congruent to:

11. ∠𝐴

12. ∠𝐹

13. ∠𝐸

Find the length represented by:

14. 𝑥

15. 𝑦

30

20

𝑥 40

2 16

𝑦

A

B C

D

E F



16. In the figure below ∆KLM~ΔNPM. Calculate the length of altitude MQ. All units are in inches.

Use the figure below answer questions 17-20.

K

L

M

N

P

Q

12

12

16

16

40

𝑏 12

8

24

𝑎

30

54°

𝜃

A

B

C

D

E



Find the length represented by:

17. 𝑎

18. 𝑏

19. Find the measure of the angle represented by 𝜃

20. Write the statement of proportionality using the ratio of corresponding sides



Answer keys

Day 73:

1. 6 ft

2. 4 in.

3. 50 ft

4. 5 in. and 6 in

5. 7.5 ft

6. 𝑥 = 16

7. 𝑥 = 6.4

8. 𝑥 = 10

9. 𝑥 = 17.5

10. 𝑥 = 18.75

11. ∠𝐷

12. ∠𝐶

13. ∠𝐵

14. 𝑥 = 12

15. 𝑦 = 8

16. 𝑀𝑄 = 30 𝑖𝑛.

17. 𝑎 = 16

18. 𝑏 = 18

19. 𝜃 = 54°

20. AD

AB=

DE

BC=

AE

AC=

5

3



In the figure below AD = 9 inches, DE = 7 inches, BC = 14 inches and DE ∥ BC.

Identify two similar triangles from the figure above and hence find the length DB.

A

B C

D E 7 in.

14 in.

9 in.



Answer keys Day 73:

ΔADE ~ ΔABC

DB = 9 inches



Identify the postulate that makes the triangles congruent from 1 – 2.

1.

2.

3. The linear scale factor of the sides of two triangles is 2

3. If one side of the bigger triangle is 18, what

would be the corresponding side of the smaller triangle?

4. The ratio between the corresponding sides of a two triangles is 𝑥+5

𝑥, where 𝑥 is the length of the side

of the smaller triangle. If the linear scale factor is 12

7, what is the length of the side corresponding to 𝑥 +

3?

5. Identify the condition(s) for a similarity transformation to exists between two figures

6. A dilation has a scale factor of 1.2. Compare the size of the image and that of the pre-image



Answer Keys

Day 74:

6. SAS

7. ASA

8. 12

9. 120

7

10. At least one rigid motion

And dilation

11. The image is larger than the pre-image by a linear scale factor factor of 1.2



1. Draw two horizontal lines of different sizes.

2. At the end points on the smaller line, draw lines inclined at 30° and 50° respectively. Connect the

lines to get the third vertex of the triangle.

3. At the end points on the larger line, draw lines inclined at 30° and 50° respectively. Connect the lines

to get the third vertex of the triangle.

4. Label the two triangles above.

5. Measure the corresponding third angles of the two triangles, what do you notice.

6. Measure the corresponding sides of the two triangles

7. Find the linear scale factor of the two triangles using each of the three pairs of corresponding sides.

8. Comment on the answers in 7 above.

9. Make a conclusion based on your answer in 8, 5 and the initial measurements given.



In this activity, students will construct similar triangles given ASA and prove that they are similar. Here

AA criterion comes on due to two A’s in ASA postulate. They work in groups of at least 4. Each group

will require a protractor, a ruler, a pencil and a plane paper.

Answer Keys

Day 74:

1-4. No response

5. They are approximately equal to 100°

6-7. Different responses

8. The linear scales are approximately equal

9. The two triangles are similar



Use the following information to answer questions 1 - 8

1. Identify two triangles in the figure above.

2. Identify corresponding sides in the two triangles above

3. Identify corresponding angles in the figure above.

4. Identify congruent angles in the two triangles stating the reason.

5. Are two triangles similar?

6. Explain your answer.

7. If yes, take 𝑘 to be the linear scale factor, write three ratios equal to 𝑘. If not, explain why 𝑘 does not

exists for the two triangles.

S V

U

W

X



Use the following information to answer the following questions 8 - 20

8. Find the size of angle DYG.

9. Identify corresponding sides between triangle DYG and DFG.

10. Identify corresponding sides between triangle DFY and DFG.

11. Identify corresponding angle between triangle DYG and DFG.

12. Identify corresponding angle between triangle DFY and DFG.

13. Identify a pair of congruent angles between triangle DYG and DFG, if any stating the reason.

14. Identify a pair of congruent angles between triangle DFY and DFG, if any stating the reason.

15. Are DYG and DFG similar, explain.

D F

G

Y

12

13 in 5 in



16. Are DFY and DFG similar? Explain.

17. Identify all pairs of congruent corresponding angles in DFY and DYG.

18. Show that angle GDY and angle YFD are equal.

19. Are the two triangles, DYG and FYD, similar?

20. Explain your answer in 19 above.



Answer keys Day 74:

1. Triangle VWU and XSU

2. UW and US, WV and SX and, VW and XU

3. ∠𝑆 and ∠𝑉𝑊𝑈, ∠𝑋 and ∠𝑊𝑉𝑈, and ∠𝑋𝑈𝑆 and ∠𝑉𝑈𝑊

4. ∠𝑆 = ∠𝑉𝑊𝑈 = 90°, and ∠𝑋𝑈𝑆 = ∠𝑉𝑈𝑊 common to both triangles

5. 𝑌𝑒𝑠

6. In 4,we have two pairs of corresponding angles equal hence by AA criterion, they are similar

7. 𝑘 =𝑈𝑊

𝑈𝑆=

𝑊𝑉

𝑆𝑋=

𝑉𝑊

𝑋𝑈

8. 90°

9. DY and FD, YG and DG; and GD and GF

10. FY and FD, YD and DG; and GF and DF

11. ∠𝐷𝑌𝐺 and ∠𝐹𝐷𝐺, ∠𝑌𝐺𝐷 and ∠𝐷𝐺𝐹; and ∠𝐺𝐷𝑌 and ∠𝐺𝐹𝐷

12. ∠𝐹𝑌𝐷 and ∠𝐹𝐷𝐺, ∠𝑌𝐷𝐹 and ∠𝐷𝐺𝐹; and ∠𝐷𝐹𝑌 and ∠𝐺𝐹𝐷

13. ∠𝐷𝑌𝐺 = ∠𝐹𝐷𝐺 = 90° (𝐺𝑖𝑣𝑒𝑛) ∠𝑌𝐺𝐷 = ∠𝐷𝐺𝐹 (Common to both triangles)

14. ∠𝐹𝑌𝐷 = ∠𝐹𝐷𝐺 = 90°(𝐺𝑖𝑣𝑒𝑛) ∠𝐷𝐹𝑌 = ∠𝐺𝐹𝐷 (Common to both triangles)

15. Yes, buy AA criterion now that ∠𝐷𝑌𝐺 = ∠𝐹𝐷𝐺 = 90° (𝐺𝑖𝑣𝑒𝑛) ∠𝑌𝐺𝐷 =

∠𝐷𝐺𝐹 (Common to both triangles)

16. Yes, buy AA criterion now that ∠𝐹𝑌𝐷 = ∠𝐹𝐷𝐺 = 90°(𝐺𝑖𝑣𝑒𝑛) ∠𝐷𝐹𝑌 = ∠𝐺𝐹𝐷 (Common to both

triangles)

17. ∠𝐷𝑌𝐺 and ∠𝐹𝑌𝐷, ∠𝑌𝐺𝐷 and ∠𝑌𝐷𝐹; and ∠𝐺𝐷𝑌 and ∠𝐷𝐹𝑌

18. Let ∠𝐷𝐹𝑌 = 𝑎, since triangle DFG is a right triangle, Let ∠𝐷𝐹𝑌 and ∠𝐷𝐺𝑌 are complementary

hence ∠𝐷𝐺𝑌 = 90 − 𝑎

Since ∠𝐷𝑌𝐺 a right angle, 𝐺𝐷𝑌 = 90 − (∠𝐷𝐺𝑌) = 90 − 90 + 𝑎 = 𝑎

Thus ∠𝐺𝐷𝑌 = 𝑎 = ∠𝐷𝐹𝑌

19. Yes

20. Now that ∠𝐺𝐷𝑌 = 𝑎 = ∠𝐷𝐹𝑌, by AA criterion, they are similar



In the figure below, PE and AB are parallel lines. Use the AA criterion to show that triangles PEG and ABG

are similar.

A B

G

P E



Answer Keys

Day 74

Since lines AB and PE are parallel, we have ∠𝑃𝐸𝐺 = ∠𝐴𝐵𝐺 and ∠𝐵𝐴𝐺 = ∠𝐸𝑃𝐺 since they are pairs

of corresponding angles. Thus, AA is achieved showing that the two triangles are similar.

146


Geometry


Week 15


147

Week 15

Weekly Assessments

148

Week #15 1. Points G and F are the midpoints of sides AB and AC respectively. a) Give two relationships between side BC and line GF. b) Find the length of GF.

2. ∆𝑀𝑁𝑂 is a dilation of ∆𝑀𝐴𝐵. a) Find the center of dilation. b) Find the length of ON.

3. Use the figure below to answer the questions that follow. a) Find the scale factor of dilation. b) Find the value of 𝑥.

4. State whether the pairs of triangles are similar or not. a) b)

A

B

C

G

F

4 𝑖𝑛

3 𝑖𝑛 2 𝑖𝑛

4 𝑖𝑛 𝑥

M

N

O

A

B

2.1 𝑖𝑛

149

5. Use the figure below to answer the questions that follow. a) Solve the following ratios.

𝐴𝐵

𝑀𝑁

𝐴𝐶

𝑂𝑁

𝐵𝐶

𝑂𝑀

b) Is ∆𝐴𝐵𝐶 ∼ ∆𝑀𝑁𝑂?

6. Use the diagram below to answer the questions that follow. a) Is ∆𝐹𝐺𝐻 ≅ ∆𝐽𝐾𝐿? b) Give a reason for your answer above.

3 𝑖𝑛

4.5 𝑖𝑛

2.25 𝑖𝑛

A B

C

2 𝑖𝑛

3 𝑖𝑛

1.5 𝑖𝑛

N M

O

G

F J

K

L H

150

Week 15 - Keys

Weekly Assessments

151

Week #15 KEY 1. Points G and F are the midpoints of sides AB and AC respectively. a) Give two relationships between side BC and line GF. 𝐺𝐹 ∥ 𝐵𝐶

𝐺𝐹 =1

2𝐵𝐶

b) Find the length of GF. 2 𝑖𝑛

2. ∆𝑀𝑁𝑂 is a dilation of ∆𝑀𝐴𝐵. a) Find the center of dilation. M b) Find the length of ON. 4.2 𝑖𝑛

3. Use the figure below to answer the questions that follow. a) Find the scale factor of dilation. 1.5 b) Find the value of 𝑥. 2 𝑖𝑛

4. State whether the pairs of triangles are similar or not. a) They are not similar b) They are similar

A

B

C

G

F

4 𝑖𝑛


4 𝑖𝑛 𝑥

M

N

O

A

B

2.1 𝑖𝑛

152

5. Use the figure below to answer the questions that follow. a) Solve the following ratios.

𝐴𝐵

𝑀𝑁= 1.5

𝐴𝐶

𝑂𝑁= 1.5

𝐵𝐶

𝑂𝑀= 1.5

b) Is ∆𝐴𝐵𝐶 ∼ ∆𝑀𝑁𝑂? Yes

6. Use the diagram below to answer the questions that follow. a) Is ∆𝐹𝐺𝐻 ≅ ∆𝐽𝐾𝐿? Yes b) Give a reason for your answer above. A reflection over a line half way between the triangles will map one triangle onto the other.

3 𝑖𝑛

4.5 𝑖𝑛

2.25 𝑖𝑛

A B

C

2 𝑖𝑛

3 𝑖𝑛

1.5 𝑖𝑛

N M

O

G

F J

K

L H



1. Use the triangle below to answer the questions that follow. 𝑄𝑅 =1

2𝐴𝐵

a) What is the ratio of AQ:QC?

b) What is the ratio of BR:RC?

c) Find the length of QR

2. In the diagram below, ∆𝑀𝑁𝑂 is similar to ∆𝐾𝐿𝑂. 𝑀𝑂 = 12𝑖𝑛, 𝐾𝑂 = 4𝑖𝑛 and 𝐿𝑂 = 6𝑖𝑛.

a) Find the length of KL

b) Find the length of LN

A B

C

𝑄 𝑅

2.6𝑖𝑛

M N

O

K L

9𝑖𝑛



Answer Key Day 76:

1. a) 1:1

b) 1:1

c) 1.3𝑖𝑛

2. a) 3𝑖𝑛

b) 12𝑖𝑛



1. Draw a 4𝑖𝑛 long line in the middle of a plane paper.

2. Label this line AB.

3. Using the method of your choice construct a line parallel to and above line AB.

4. Make a mark anywhere above the line you have constructed and label it as C.

5. Using a ruler and a pencil join point C and end A.

6. Using a ruler and a pencil join point C and end B such that you have ∆𝐴𝐵𝐶 and a line parallel to side

AB passing through it.

7. Label the points where the parallel line intersects side AC and BC as Q and R respectively.

8. Using a ruler measure the lengths of AQ, QC, BR and RC and record them in the table below.

Line

AQ QC BR RC

Length

9. The ratios 𝐴𝑄

𝑄𝐶 and

𝐵𝑅

𝑅𝐶. Is 𝐴𝑄

𝑄𝐶=

𝐵𝑅

𝑅𝐶?



In this activity students will draw a triangle of their choice and a line parallel to one of the sides then

establish the proportionality of the parts of the sides divided by the parallel line.

Students will work in groups of at least three and each group is required to have a pencil, a ruler a

compass and a plane paper.

Answer Keys

Day 76:

1-8. No response

9. Yes



Use the figure below to answer questions 1 and 4.

𝐴𝐶 = 11𝑖𝑛, 𝐴𝐷 = 5𝑖𝑛, 𝐴𝐹 = 4𝑖𝑛 and 𝐶𝐸 = 5𝑖𝑛.

1. Find the length of FB

2. Find the length of AB

3. Find the length of EB?

4. Find the length of BC

Use the diagram below to answer questions 5 and 8.

5. Find the length of AK

A B

C

D E

F

12𝑖𝑛

4𝑖𝑛

J 6𝑖𝑛 A K

B

L

C

5in



6. Find the length of JK

7. Find the length of CL

8. Find the length of KL

Use the figure below to answer questions 9 and 11.

9. In which ratio does D divide SU?

10. Find the value of x

11. What is the length of side SU?

Use the diagram below to answer questions 12-17. 𝐴𝐺 = 12𝑖𝑛, 𝐺𝐹 = 8𝑖𝑛, 𝐴𝐵 = 8𝑖𝑛, 𝐶𝐷 = 9𝑖𝑛, 𝐷𝐸 =

10𝑖𝑛, 𝑂𝐵 = 3𝑖𝑛, 𝑂𝐶 = 2𝑖𝑛 and 𝑂𝐸 = 8𝑖𝑛

12. Find the length of AC

S 2𝑖𝑛 𝐸 4𝑖𝑛 T

D

U

5𝑖𝑛

𝑥

𝐴 𝐵 𝐶 𝐷

G E

F

O



13. Find the length of BC

14. Find the length of AB

15. Find the length of FE?

16. Find the length of DF?

17. Find the length of CG?

Use the diagram below to questions 18 - 19

18. Find the value of 𝑥

19. What is the length of MN?

20. Find the value of y in the figure below.

𝑀 𝑥 𝑅 15𝑖𝑛 𝑁

𝑄

O

24𝑖𝑛

18𝑖𝑛

y

9𝑖𝑛 15𝑖𝑛

3𝑖𝑛



Answer Key

1. 4.8𝑖𝑛

2. 8.8𝑖𝑛

3. 4.17𝑖𝑛

4. 9.17𝑖𝑛

5. 2𝑖𝑛

6. 8𝑖𝑛

7. 15𝑖𝑛

8. 20𝑖𝑛

9. 1:2

10. 2.5𝑖𝑛

11. 7.5 𝑖𝑛

12. 13.5𝑖𝑛

13. 3.375𝑖𝑛

14. 10.125

15. 8.18𝑖𝑛

16. 18.18𝑖𝑛

17. 18𝑖𝑛

18. 11.25𝑖𝑛

19.26.25𝑖𝑛

20.1.8𝑖𝑛



Use the diagram below to answer the question that follow.

1. Find the value of 𝑥

𝑥 2.5𝑖𝑛

10𝑖𝑛 8𝑖𝑛



Answer Keys

Day 76:

1. 2𝑖𝑛



Use the figure below to answer the following questions. All length measurements are given in inches.

1. (a) Identify two triangles similar to ∆ABC.

(b) Identify two triangles that share 𝛼 as one of their angles.

(c) Identify two triangles that share 𝛽 as one of their angles.

2. Using the similar triangles you have identified in the figure above and the ratio of corresponding sides

for proportionality, calculate to two decimal places, the length of the following sides:

(a) BC

(b) BD

A

B C

D

𝛼

𝛽

16

8

24



Answer keys Day 77:

1. (a) ∆BDC and ∆ADB

(b) ∆ABC and ∆ADB

(c) ∆ABC and ∆BDC

2. (a) BC = 27.71 in.

(b) BD = 13.86 in.



1. Use suitable measurements to construct right ΔABC using a ruler and a protractor on the blank paper

such that ∠BAC = 60°, ∠ABC = 90° and ∠ACB = 30°. ΔABC should appear as show below.

2. Drop a perpendicular from point B to intersect AC̅̅̅̅ at point D as shown below.

3. Identify triangles ΔBDC and ΔADB from ΔABC and sketch them on the blank paper. They should

appear as shown below.

A

B C

A

B C

D

A

D B

B

D C



4. Measure ∠ADB and ∠BDC and compare their measures to ∠ABC. What do you notice?

5. Identify two triangles that have ∠A as one of their angles.

6. Identify two triangles that have ∠C as one of their angles.

7. Considering the shapes and sizes of the angles of the triangles above, give the major relationship

between the three triangles above?

8. Measure the lengths AB̅̅ ̅̅ , BC̅̅̅̅ , AD̅̅ ̅̅ , DC̅̅̅̅ , BD̅̅ ̅̅ and AC̅̅̅̅ in inches.

9. Find the sum of the squares of the lengths AB̅̅ ̅̅ and BC̅̅̅̅ and compare it to the square of the length AC̅̅̅̅

on ∠ABC . Write down an identity to show the relationship between the three sides.

10. Find the sum of the squares of the lengths AD̅̅ ̅̅ and BD̅̅ ̅̅ and compare it to the square of the length AB̅̅ ̅̅

on ∠ADB . Write down an identity to show the relationship between the three sides.

11. Find the sum of the squares of the lengths BD̅̅ ̅̅ and CD̅̅̅̅ and compare it to the square of the length BC̅̅̅̅

on ∠BDC . Write down an identity to show the relationship between the three sides.



In this activity students will work in groups of four to verify the Pythagoras theorem from similar right

triangles. The students in the respective groups will require a ruler, blank paper and a protractor.

Answer keys Day 77:

1. No response

2. No response

3. No response

4. ∠ADB = ∠BDC = ∠ABC = 90°; the three angles are congruent

5. ΔABC and ΔADB

6. ΔABC and ΔBDC

7.The three triangles are similar

8. The lengths should be accurately measured

9. AB̅̅ ̅̅ 2 + BC̅̅̅̅ 2 = AC̅̅̅̅ 2

10. AD̅̅ ̅̅ 2 + BD̅̅ ̅̅ 2 = AB̅̅ ̅̅ 2

11. BD̅̅ ̅̅ 2 + CD̅̅̅̅ 2 = BC̅̅̅̅ 2



In the figure 𝑃𝑄 = 𝑟, 𝑄𝑅 = 𝑝, 𝑃𝑅 = 𝑞, 𝑃𝑆 = 𝑛, 𝑅𝑆 = 𝑚 and 𝑄𝑆 ⊥ 𝑃𝑅 at 𝑆 . Study it and use it to answer

questions 1-8 below.

1. Express 𝑞 in terms of 𝑛 and 𝑚.

Given that ∠QPS = 56° and ∠PRQ = 34°. Find the measures of the following angles:

2. ∠QSR

3. ∠SQR

4. ∠PSQ

5. ∠PQS

6. Identify two triangles similar to ΔPQR and label them in such a way that the corresponding parts

match the parts of ΔPQR.

P

Q R

S

𝑛

𝑚 𝑟

𝑝

𝑞 56°

34°



Complete the proportionality statements represented below:

7. 𝑝

=𝑞

𝑝

8. 𝑟

𝑛=

𝑟

Use the proportionality statements in questions 7 and 8 to complete the equations below:

9. 𝑝2 = 𝑞 × ____

10. 𝑟2 = 𝑛 × ____

11. Use the equations in questions 9 and 10 to show that 𝑝2 + 𝑟2 = 𝑞2 by substitution.

In the figure below, ΔABC is a right triangle and BD ⊥ AC. Use it to answer questions 12-20.

A

C B

D

𝑐

𝑦

𝑥

𝑎

𝑏

47°

43°



Given that ∠CBD = 47° and ∠ABD = 43°. Calculate the measures of the following angles:

12. ∠BCD

13. ∠BAD

14. ∠ADB

15. Write 𝑏 in terms of 𝑥 and 𝑦

Fill in the gaps to complete the proportionality statements below:

16. 𝑎

𝑥=

𝑎 with reference to ΔABC and ΔBDC

17. 𝑐

𝑦=

𝑐 with reference to ΔABC and ΔADB

Use the proportionality statements in questions 16 and 17 to complete the equations below:

18. 𝑎2 = 𝑥 × ____

19. 𝑐2 = 𝑦 × ____

20. Use the equations in questions 18 and 19 prove the identity 𝑎2 + 𝑐2 = 𝑏2 by substitution.



Answer keys Day 77:

1. 𝑞 = 𝑚 + 𝑛

2. 90°

3. 56°

4. 90°

5. 34°

6. ΔPSQ and ΔQSR

7. 𝑝

𝑚=

𝑞

𝑝

8. 𝑟

𝑛=

𝑞

𝑟

9. 𝑝2 = 𝑞 × 𝑚

10. 𝑟2 = 𝑛 × 𝑞

11. 𝑝2 + 𝑟2 = 𝑞𝑚 + 𝑞𝑛 = 𝑞(𝑚 + 𝑛) but 𝑚 + 𝑛 = 𝑞 hence 𝑞(𝑚 + 𝑛) = 𝑞 × 𝑞 = 𝑞2

∴ 𝑝2 + 𝑟2 = 𝑞2

12. 43°

13. 47°

14. 90°

15. 𝑏 = 𝑥 + 𝑦

16. 𝑎

𝑥=

𝑏

𝑎

17. 𝑐

𝑦=

𝑏

𝑐

18. 𝑎2 = 𝑥 × 𝑏

19. 𝑐2 = 𝑦 × 𝑏

20. 𝑎2 + 𝑐2 = 𝑏𝑥 + 𝑏𝑦 = 𝑏(𝑥 + 𝑦) but 𝑥 + 𝑦 = 𝑏 hence 𝑏(𝑥 + 𝑦) = 𝑏 × 𝑏 = 𝑏2

∴ 𝑎2 + 𝑐2 = 𝑏2



In the figure 𝐴𝐵 = 𝑐, 𝐵𝐶 = 𝑎, 𝐴𝐶 = 𝑏, 𝐴𝐷 = 𝑥 and 𝐷𝐶 = 𝑦.

(a) Write 𝑏 in terms of 𝑥 and 𝑦.

(b) Complete the proportionality statement below using the sides on ΔABC and ΔADB.

𝑐=

𝑏

𝑐

(c) Complete the proportionality statement below using the sides on ΔABC and ΔBDC.

𝑎=

𝑏

𝑎

A

B C

D

𝑥

𝑦 𝑐

𝑎

𝑏



Answer keys Day 77:

(a) 𝑏 = 𝑥 + 𝑦

(b) 𝑐

𝑥=

𝑏

𝑐

(c) 𝑎

𝑦=

𝑏

𝑎



1. Use triangles below to answer the questions that follow.

a) Which criterion makes the two triangles similar?

b) Find the value of y

c) Find the value of x


a) Which postulate makes the triangles above congruent?

b) Find the value of z


15 𝑖𝑛 12 𝑖𝑛

𝑥 𝑦

(3𝑧 − 2) 𝑖𝑛

7 𝑖𝑛



Answer Key Day 78:

1. a) AA criterion

b) 5 𝑖𝑛

c) 4 𝑖𝑛

2 a) A.S.A postulate

b) 3



1. Draw a rectangle measuring 6 in by 4 in on a plane paper and label it ABCD as shown.

2. Mark the midpoint of AB and label it as O.

3. Draw a straight line joining point O and C.

4. Join points D and O with a straight line.

5. Measure the lengths of sides DO and OC.

Are they equal?

6. Measure the lengths of AO and OB.

Are they equal?

7. Measure the lengths of AD and CB.

Is ∆𝐴𝐷𝑂 ≅ ∆𝐵𝐶𝑂?

Explain your answer.

D C

A B



In this activity students will draw different triangles and identify the ones that are congruent. Students

will work in groups of at least three and each group is required to have a ruler, a pencil and a plane

paper.

Answer Keys

Day 78:

1-4. No response

5. Yes

6. Yes

7. Yes, S.S.S postulate



Use the diagram below to answer the questions 1-3.

1. Find the value of r

2. Find the value of p

3. Find the length of AC

4. A student who is 5ft has a shadow of length 15ft. At the same time, the length of the shadow of a

building was 450 ft. What is the height of the building?

Use the diagram below to answer the questions 5 and 6.

5. Find the value of y

2.4 𝑖𝑛 0.8 𝑖𝑛

𝑦

1.1 𝑖𝑛 𝑧

1.8 𝑖𝑛

𝐴 6 𝑖𝑛 𝐵 𝑟 𝐶

3 𝑖𝑛

12 𝑖𝑛

𝑝

9 𝑖𝑛

E

D



6. Find the value of z

Use the diagram below to answer questions 7-11.

𝑀𝑁 = 12𝑖𝑛.

7. Find the value of 𝑥.

8. Find the value of 𝑡

9. Find the value of 𝑦

10. Find the value of s

11. What is the value of w?

12. A mobile phone manufacturing company makes two rectangular models of mobile phones such that

they are similar. The first model has a width of 2 in and a length of 3 in. If the second model has a width

of 2.5 in, what is its length?

A B M N

D E

C O

12 𝑖𝑛

8 𝑖𝑛

10 𝑖𝑛

𝑡 3 𝑖𝑛

𝑠

(5𝑥 + 2) 𝑖𝑛

y 𝑤




13. What is the value of 𝑑?

14. Find the value of b

15. Find the value of c

Use the diagram below to answer questions 16 to 20.

The two triangles are image and pre-image of one another under glide reflection.

16. What is the value of 𝑗?

(2𝑏 − 2)

(4𝑐) 𝑖𝑛

10 𝑖𝑛

6 𝑖𝑛

(25 + 2𝑑)°

(3𝑑 − 15)°

2 𝑖𝑛

8 𝑖𝑛

𝑘 1.5 𝑖𝑛

𝑙

9 𝑖𝑛 m

𝑖

𝑗



17. Find the value of 𝑘

18. Find the value of l

19. Find the value of the of m

20. Find the value of 𝑖



Answer Keys

Day 78:

1. 2 𝑖𝑛

2. 3 𝑖𝑛

3. 8 𝑖𝑛

4. 150 𝑓𝑡

5. 3.3 𝑖𝑛

6. 0.6 𝑖𝑛

7. 𝑥 = 2 𝑖𝑛

8. 𝑡 = 5 𝑖𝑛

9. 15 𝑖𝑛

10. 6 𝑖𝑛

11. 9 𝑖𝑛

12. 3.75 𝑖𝑛

13. 40

14. 6

15. 3

2

16. 8 𝑖𝑛

17. 6 𝑖𝑛

18. 3 𝑖𝑛

19. 12 𝑖𝑛

20. 7.5 𝑖𝑛



1. Find the value of 𝑎 in the diagram below.

(𝑎 + 2) 7 𝑖𝑛



Answer Keys Day 78:

1. 5 𝑖𝑛



Use the following diagram to answer the following questions if 𝐺𝑇̅̅ ̅̅ is parallel to 𝐻𝐾̅̅ ̅̅ .

1. Identify two triangle from the diagram above

2. Are the triangles similar or not?

3. Explain your answer.

4. Which condition should the two parallel lines meet for WHKT to be a parallelogram?

5. State the condition that should be met for the two triangles to be congruent.

.

G H

J

K

T



Answer Keys

Day 79:

1. ∆𝑇𝐽𝐺 and ∆𝐾𝐽𝐻

2. Yes

3. Corresponding angles are equal

∠𝑇𝐺𝐽 = ∠𝐾𝐻𝐽(Corresponding angles)

∠𝐽𝑇𝐺 = ∠𝐽𝐾𝐻(Corresponding angles)

∠𝐺𝐽𝑇 = ∠𝐻𝐽𝐾(Common to both triangles)

4. 𝐺𝑇̅̅ ̅̅ = 2𝐻𝐾̅̅ ̅̅ .

5. 𝐺𝑇̅̅ ̅̅ = 𝐻𝐾̅̅ ̅̅ .

G H

J

K

T



1. Draw a rectangle LMNO using a ruler, a pencil and a protractor.

2. Draw a diagonal from L to N.

3. Draw another diagonal from M to O.

4. Label the intersection of the diagonals as P.

5. Measure LM and MO. What do you realize?

6. Measure LP and MP

7. Write a relation between LP and LN.

8. Write a relation between MO and MP.

9. Make a conclusion based on your answer in 8 and 7 above.



In this activity, students will show that the diagonals of a rectangle bisect each other at the point of

intersection. This is taken as a verification of the proof in the presentation. They will work in groups of at

least 3. Each group will require a protractor, a ruler, a pencil and a plane paper.

Answer Keys

Day 79:

1-4. No response

5. Difference responses

They are approximately equal

6. Different responses

7. 2𝐿𝑃 = 𝐿𝑁

8. 2𝑀𝑃 = 𝑀𝑂

9. The diagonals are bisected at the intersection, P.




Consider the rhombus below. We want to proof that the diagonals of a rhombus bisects the angles at

their endpoints and intersect at a right angle.

Statement Reason

𝐺𝑆̅̅̅̅ = 𝑆𝑇̅̅̅̅ = 𝑇𝐻̅̅ ̅̅ = 𝐻𝐺̅̅ ̅̅ 1.

In triangle GHT and GST, 𝐺𝑇 = 𝐺𝑇 2.

𝑆𝑇 = 𝐻𝑇, 𝐺𝑆 = 𝐺𝐻, 3.

Triangles GHT and GST are congruent 4.

∠𝐻𝐺𝑇 = ∠𝑆𝐺𝑇, ∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 5.

∠𝑆𝑇𝐻 = ∠𝑆𝑇𝐺 + ∠𝐺𝑇𝐻; ∠𝑆𝐺𝐻 = ∠𝑆𝐺𝑇 +

∠𝑇𝐺𝐻

6.

∠𝑆𝑇𝐻 = 2∠𝑆𝑇𝐺 = 2∠𝐺𝑇𝐻; ∠𝑆𝐺𝐻 = 2∠𝑆𝐺𝑇 =

2∠𝑇𝐺𝐻

From 5 and 6 above

Diagonals of a rhombus intersects each other 7.

G H

T S

O



Let ∠𝐺𝑇𝐻 = 𝛼

∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 = 𝛼 8.

𝐺𝐻 ∥ 𝑆𝑇 and 𝐺𝑆 ∥ 𝐻𝑇 9.

∠𝐻𝐺𝑇 = ∠𝑆𝑇𝐺 10.

∠𝑆𝑇𝐺 = ∠𝐻𝐺𝑇 = 𝛼; ∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 = 𝛼 11.

∠𝑆𝑇𝐻 = 2𝛼; ∠𝑆𝐺𝐻 = 2𝛼 13.

∠𝐺𝑆𝑇 + ∠𝑆𝐺𝐻 = 180° 14.

∠𝐺𝑆𝑇 + 2𝛼 = 180° 15.

∠𝐺𝑆𝑇 = 180° − 2𝛼 16.

∠𝐺𝐻𝑇 = ∠𝐺𝑆𝑇 = 180° − 2𝛼 17.

∠𝐺𝑆𝐻 = ∠𝑇𝑆𝐻 = 90° − 𝛼; ∠𝐺𝐻𝑆 = ∠𝑆𝐻𝑇

= 90° − 𝛼

18.

∠𝐻𝑂𝑇 = ∠𝑇𝑂𝑆 = ∠𝑆𝑂𝐺 = ∠𝐺𝑂𝐻

= 180 − ((90 − 𝛼) + 𝛼) = 90°

19.

Diagonals of a rhombus intersects at right angle 20.



Answer keys Day 79:

Statement Reason

𝐺𝑆̅̅̅̅ = 𝑆𝑇̅̅̅̅ = 𝑇𝐻̅̅ ̅̅ = 𝐻𝐺̅̅ ̅̅ 1. Properties of rhombus

In triangle GHT and GST, 𝐺𝑇 = 𝐺𝑇 2. Common to both triangles

𝑆𝑇 = 𝐻𝑇, 𝐺𝑆 = 𝐺𝐻, 3. By properties of a rhombus

Triangles GHT and GST are congruent 4.Corresponding sides are equal

∠𝐻𝐺𝑇 = ∠𝑆𝐺𝑇, ∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 5.Corresponding angles of congruent triangles

∠𝑆𝑇𝐻 = ∠𝑆𝑇𝐺 + ∠𝐺𝑇𝐻; ∠𝑆𝐺𝐻 = ∠𝑆𝐺𝑇 +

∠𝑇𝐺𝐻

6. Sum of Adjacent angles

∠𝑆𝑇𝐻 = 2∠𝑆𝑇𝐺 = 2∠𝐺𝑇𝐻; ∠𝑆𝐺𝐻 = 2∠𝑆𝐺𝑇 =

2∠𝑇𝐺𝐻

From 5 and 6 above

Diagonals of a rhombus intersects each other 7.∠𝑆𝑇𝐻 = 2∠𝑆𝑇𝐺 = 2∠𝐺𝑇𝐻; ∠𝑆𝐺𝐻 =

2∠𝑆𝐺𝑇 = 2∠𝑇𝐺𝐻



Let ∠𝐺𝑇𝐻 = 𝛼

∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 = 𝛼 8.Since ∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺

𝐺𝐻 ∥ 𝑆𝑇 and 𝐺𝑆 ∥ 𝐻𝑇 9. Properties of rhombus

∠𝐻𝐺𝑇 = ∠𝑆𝑇𝐺 10. Alternate angles

∠𝑆𝑇𝐺 = ∠𝐻𝐺𝑇 = 𝛼; ∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 = 𝛼 11. Since ∠𝐻𝐺𝑇 = ∠𝐺𝑇𝑆 and ∠𝐻𝐺𝑇 = 𝛼

Since ∠𝐻𝑇𝐺 = ∠𝑆𝑇𝐺 and ∠𝐻𝑇𝐺 = 𝛼

∠𝑆𝑇𝐻 = 2𝛼; ∠𝑆𝐺𝐻 = 2𝛼 13. ∠𝑆𝑇𝐻 = 2∠𝑆𝑇𝐺 = 2∠𝐺𝑇𝐻; ∠𝑆𝐺𝐻 =

2∠𝑆𝐺𝑇 = 2∠𝑇𝐺𝐻

∠𝐺𝑆𝑇 + ∠𝑆𝐺𝐻 = 180° 14. Adjacent angles of a rhombus

∠𝐺𝑆𝑇 + 2𝛼 = 180° 15. Substitution; ∠𝑆𝐺𝐻 = 2𝛼

∠𝐺𝑆𝑇 = 180° − 2𝛼 16. Algebraic equality of substitution

∠𝐺𝐻𝑇 = ∠𝐺𝑆𝑇 = 180° − 2𝛼 17. Opposite angles of a rhombus

∠𝐺𝑆𝐻 = ∠𝑇𝑆𝐻 = 90° − 𝛼; ∠𝐺𝐻𝑆 = ∠𝑆𝐻𝑇

= 90° − 𝛼

18. Diagonals of a rhombus intersects each other

∠𝐻𝑂𝑇 = ∠𝑇𝑂𝑆 = ∠𝑆𝑂𝐺 = ∠𝐺𝑂𝐻

= 180 − ((90 − 𝛼) + 𝛼) = 90°

19. Interior angles of a triangle

Diagonals of a rhombus intersects at right angle 20. ∠𝐻𝑂𝑇 = ∠𝑇𝑂𝑆 = ∠𝑆𝑂𝐺 = ∠𝐺𝑂𝐻 = 90°



Determine if the two triangles in the figure below are similar if angle DFH and HGE are equal. Explain

your answer.

D E

F

G

H



Answer Keys

Day 79

∠𝐷𝐹𝐻 and ∠𝐻𝐺𝐸 are equal (Given)

∠𝐺𝐸𝐻 = ∠𝐹𝐸𝐷 (Common to both triangles)

Thus AA criteria is satisfied showing that they are similar

195


Geometry


Week 16


196

Week 16

Weekly Assessments

197

Week #16 1. Use the diagram below to answer the questions that follow. a) Are the two triangles similar? b) Give a reason for your answer in (a).

2. State whether each pair of figures are similar or not. a) b)

3. In the triangle below, Points V and Q are midpoints of SU and ST respectively. ∠𝑉𝑇𝑈 =29°. a) State two relationships between line VQ and TU. b) Find the size of ∠𝑉𝑇𝑈.

4. Use the figure below to answer the questions that follow. a) Find the value of 𝑥. b) Find the value of 𝑦.

37°

37°

4 𝑖𝑛

4 𝑖𝑛

2 𝑖𝑛

3 𝑖𝑛

S T

U

V

Q

(2𝑥 + 1) 𝑖𝑛

(4𝑥 + 2) 𝑖𝑛

4 𝑖𝑛

2 𝑖𝑛

𝑦

12 𝑖𝑛

198

5. Use the triangle below to answer the questions that follow. a) Which angle corresponds to ∠𝐾𝑄𝐵? b) Which angle corresponds to ∠𝐴𝐾𝑄? c) Find the length of BJ.

6. Use the figure below to answer the

questions that follow. 𝐴𝐽

𝐵𝐽=

2

3.

a) Find the ratio 𝐶𝐾

𝐴𝐾.

b) Find the ratio 𝐽𝐾

𝐵𝐶.

J K

Q

A 3 𝑖𝑛 2 𝑖𝑛

3.5 𝑖𝑛

B J K

A

B C

199

Week 16 - Keys

Weekly Assessments

200

Week #16 KEY 1. Use the diagram below to answer the questions that follow. a) Are the two triangles similar? Yes b) Give a reason for your answer in (a). Two corresponding angles are congruent(AA criteria is achieved)

2. State whether each pair of figures are similar or not. a) They are not similar b) They are similar

3. In the triangle below, Points V and Q are midpoints of SU and ST respectively. ∠𝑉𝑇𝑈 =29°. a) State two relationships between line VQ and TU.

𝑉𝑄 ∥ 𝑇𝑈

𝑉𝑄 =1

2𝑇𝑈

b) Find the size of ∠𝑄𝑉𝑇.

29°

4. Use the figure below to answer the questions that follow. a) Find the value of 𝑥. 2 b) Find the value of 𝑦. 1 𝑖𝑛

37°

37°

4 𝑖𝑛

4 𝑖𝑛

2 𝑖𝑛

3 𝑖𝑛

S T

U

V

Q

(2𝑥 + 1) 𝑖𝑛

(3𝑥 + 4) 𝑖𝑛

4 𝑖𝑛

2 𝑖𝑛

𝑦

12 𝑖𝑛

201

5. Use the triangle below to answer the questions that follow. a) Which angle corresponds to ∠𝐾𝑄𝐵? ∠𝐴𝐵𝐽 b) Which angle corresponds to ∠𝐴𝐾𝑄? ∠𝐽𝐴𝐵 c) Find the length of BJ. 2.33 𝑖𝑛

6. Use the figure below to answer the

questions that follow. 𝐴𝐽

𝐵𝐽=

2

3.

a) Find the ratio 𝐶𝐾

𝐴𝐾.

3

2

b) Find the ratio 𝐽𝐾

𝐵𝐶.

2

5

J K

Q

A 3 𝑖𝑛 2 𝑖𝑛

3.5 𝑖𝑛

B J K

A

B C



1. Use the triangle below to answer the questions that follow.

a) Find the value of 𝑥

b) What is the length of side JL?


a) What is the value of y?

b) Which angle corresponds to ∠𝐴

c) Which angle corresponds to ∠𝐵

18 𝑖𝑛

𝑥

15 𝑖𝑛

10 𝑖𝑛

J K

L

A B

N M

C

2 𝑖𝑛 𝑦




Answer Key

Day 81:

1. a) 12 𝑖𝑛

b) 30 𝑖𝑛

2. a) 3 𝑖𝑛

b) ∠𝐶𝑀𝑁

c) ∠𝐶𝑁𝑀



1. Draw a straight line of length 3 𝑖𝑛 in the middle of a plane paper and label it AB.

2. Mark an arbitrary point 4 𝑖𝑛 above line AB and label it C.

3. Using a ruler and a pencil, join end A and point C.

4. Using a ruler and a pencil, join end B point C to make ∆𝐴𝐵𝐶.

5. Construct a line parallel to side AB and above side AB which intersects side AC and BC.

6. Label the points where the line you have constructed intersects sides AC and BC as D and E

respectively.

7. Using a protractor measure the size of ∠𝐶𝐴𝐵 and the size of ∠𝐶𝐷𝐸.

Is ∠𝐶𝐴𝐵 = ∠𝐶𝐷𝐸?

8. Using a protractor measure the size of ∠𝐴𝐵𝐶 and the size of ∠𝐷𝐸𝐶.

Is ∠𝐴𝐵𝐶 = ∠𝐷𝐸𝐶?

9. Is ∆𝐴𝐵𝐶~∆𝐷𝐸𝐶? Explain your answer.



In this activity students will draw a triangle, construct a line parallel to one side and use AA criterion to

check whether the resulting triangles are similar. Students will work in groups of at least three and each

group is required to have a plane paper, a ruler, a protractor and a pencil.

Answer Keys

Day 81:

1-6. No response

7. Yes

8. Yes

9. Yes. Two corresponding pairs of angles are congruent.



1. Use the diagram below to answer the questions 1-5.

1. Find the ratio 𝐴𝑅

𝑅𝐶

2. Find the ratio 𝐴𝑄

𝑄𝐵


𝐴𝐶

4. Find the ratio 𝐴𝑄

𝐴𝐵

5. State whether ∆𝐴𝐵𝐶~∆𝐴𝑄𝑅? Explain your answer

1.5 𝑖𝑛 𝑄

2 𝑖𝑛

8 𝑖𝑛

6 𝑖𝑛 A B

C

R



Use the figure below to answer questions 6 -10.

6. Find the ratio 𝑂𝐴

𝐴𝑀

7. Find the ratio 𝑂𝐵

𝐵𝑁

8. Find the ratio 𝑂𝐴

𝑂𝑀

9. Find the ratio 𝑂𝐵

𝑂𝑁

10. State whether ∆𝐴𝐵𝑂~∆𝑀𝑁𝑂. Explain your answer

M N

O

𝐴 9 𝑖𝑛 B

7.2 𝑖𝑛

12 𝑖𝑛

14 𝑖𝑛

231

3 𝑖𝑛



Use the diagram below to answer questions 11-15

11. Find the ratio 𝐽𝑄

𝑄𝐾

12. Find the ratio 𝐽𝑃

𝑃𝐿

13. Find the ratio 𝐽𝑄

𝐽𝐾

14. Find the ratio 𝐽𝑃

𝐽𝐿

15. State whether ∆𝐽𝑄𝑃~∆𝐽𝐾𝐿. Explain your answer.

K

L

J

3 𝑖𝑛

P

Q

6 𝑖𝑛





𝑆𝑇 = 21 𝑖𝑛, 𝐶𝐵 = 7 𝑖𝑛, 𝐵𝑇 = 18 𝑖𝑛 𝑎𝑛𝑑 𝐶𝑈 = 6 𝑖𝑛

16. Find the ratio 𝑆𝐴

𝐴𝑇

17. Find the ratio 𝑈𝐵

𝐵𝑇

18. Find the ratio 𝑆𝐴

𝑆𝑇

19. Find the ratio 𝑈𝐵

𝑈𝑇

20. State whether ∆𝑆𝑇𝑈~∆𝐴𝑇𝐵. Explain your answer.

12 in

S

T

U

A

B

10 𝑖𝑛

20 𝑖𝑛

24 𝑖𝑛



Answer Keys

Day 81:

1. 1

4

2. 1

4

3. 1

5

4. 1

5

5. ∆𝐴𝐵𝐶~∆𝐴𝑄𝑅. The corresponding sides are proportional and the included angle is shared.

6. 3

5

7. 3

5

8. 3

8

9. 3

8

10. ∆𝐴𝐵𝑂~∆𝑀𝑁𝑂. The corresponding sides are proportional and the included angle is shared.

11. 1

2

12. 1

2

13. 1

3

14. 1

3

15. ∆𝐴𝐵𝑂~∆𝑀𝑁𝑂. The corresponding sides are proportional and the included angle is shared.

16. 5

12

17. 3

5

18. 5

17

19. 3

8

20. ∆𝑆𝑇𝑈 is not congruent to ∆𝐴𝑇𝐵. The corresponding sides are not proportional.



1. Find the value of 𝑎 in the figure below.

36 𝑖𝑛

12 𝑖𝑛



Answer Keys

Day 81:

1. 28 𝑖𝑛



1. In the figure below, 𝐴𝐵 = 𝑐, 𝐵𝐶 = 𝑎, 𝐴𝐶 = 𝑏, 𝐴𝐷 = 𝑥 and 𝐵𝐷 ⊥ 𝐴𝐶. Use it to answer the questions

below.

(a) Write an equation in terms 𝑎, 𝑏 and 𝑐 to show the relationship between the three sides of ΔABC

using the Pythagorean theorem.

(b) Find the length CD̅̅̅̅ in terms of 𝑏 and 𝑥.

(c) Given that BD̅̅ ̅̅ = 7 inches and CD̅̅̅̅ = 24 inches, calculate the lengthBC̅̅̅̅ .

2. Factorize the expression 𝑘2 − 𝑙2

3. Express 𝑙 in terms of 𝑎 and 𝑝 in the statement of proportionality below.

𝑙

𝑎=

𝑝

𝑙

A

B C

D

𝑐

𝑏

𝑥

𝑎



Answer keys

Day 82:

1. (a) 𝑎2 + 𝑐2 = 𝑏2

(b) 𝑏 − 𝑥

(c) 25 inches.

2. (𝑘 + 𝑙)(𝑘 − 𝑙)

3. 𝑙 = ±√𝑎𝑝



1. Use a ruler and a protractor to construct right ΔABC such that ∠BAC = 60°, ∠ABC = 90° and

∠ACB = 30°. Label the triangle as shown below.

2. Drop a perpendicular from point B to intersect AC̅̅̅̅ at point D as shown below.

3. Identify triangles ΔBDC and ΔADB from ΔABC, measure all their sides and note them down.

4. Verify that ΔBDC has sides related by the relation:

(𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔)2 + (𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔)2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒2

A

B C

A

B C

D



5. Verify that ΔADB has sides related by the relation:

(𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔)2 + (𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔)2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒2

6. Use the measurements in question 3 above for ΔBDC and ΔADB to find the ratio of the lengths 𝐷𝐶

𝐷𝐵

and compare it to 𝐵𝐷

𝐴𝐷. What do you notice?

7. In relation to your response to question 6 above, state whether ΔBDC~ΔADB or not. If they are

similar, state the similarity criterion you have used.



In this activity students will work in small groups of two or three to discover how the Pythagorean

Theorem yields similarity in right triangles. Students will require a ruler, a protractor and a pair of

compasses.

Answer keys

Day 82:

1. No response

2. No response

3. Emphasize on accuracy

4. (𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔)2 + (𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔)2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒2 holds

5. (𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔)2 + (𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔)2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒2 holds

6. 𝐷𝐶

𝐷𝐵 =

𝐵𝐷

𝐴𝐷

7. ΔBDC~ΔADB.

Side-Angle-Side (S.A.S) Similarity criterion



In the figure below O is the center of the circle, 𝐾𝑀 = 𝑝, 𝑀𝑂 = 𝑞, 𝑂𝐾 = 𝑂𝑁 = 𝑟 and 𝐿𝑁 is the

diameter. Use it to answer questions 1-10.

Write the following lengths in terms of 𝑝, 𝑞 and 𝑟:

1. OL̅̅̅̅

2. MN̅̅̅̅̅

3. LM̅̅ ̅̅

4. ΔKMO is a right triangle and therefore the Pythagorean theorem holds. Write 𝑝2in terms of 𝑞2 and

𝑟2.

N

K

L M O

𝑟

𝑟 𝑞

𝑝



The following equations that lead to the proportional statement.

5. Use the equation you have written in 4 above and a difference two squares to 𝑝2 in terms of two

factors.

6. Divide both sides of the equation in 5 above by 𝑝 then write an expression for 𝑝.

7. Divide the equation you have obtained in 6 above by a suitable factor to obtain a proportion

similar to the one below hence fill in the blank spaces.

𝑝

________=

𝑟+𝑞

_________

In each case, identify the sides on ΔKML that correspond to the following sides on ΔKMN.

8. KM̅̅̅̅̅

9. MN̅̅̅̅̅

10. Both ΔKML~ΔKMN are right angles. Comparing the lengths of corresponding sides you have

identified in question 8 and 9 above to the proportional statement in question 7, state whether

ΔKML~ΔKMN or not.



In right 𝚫𝐀𝐁𝐂 below = 𝒌, 𝑩𝑪 = 𝒎, 𝑨𝑪 = 𝒍, 𝑩𝑫 = 𝒑 𝒂𝒏𝒅 𝑫𝑪 = 𝒏 . Use it to answer questions 11-20.

11. Express 𝑚 in terms of 𝑝 and 𝑛.

12. Write an equation in terms of 𝑘, 𝑙 and 𝑚 to show that Pythagoras’ theorem holds for right ΔABC.

13. Express the ratio in which D divides BC in terms of 𝑝 and 𝑛.

The equations below which lead to the proportionality statement.

14. Given that D divides BC in terms of 𝑝 and 𝑛 in a ratio in question 13 above, form an equation in

terms of 𝑚, 𝑛 and 𝑝 and hence complete the equation below by filling in the missing expression.

𝑚𝑝

𝑛+𝑝+

𝑛+𝑝= 𝑚

15. By factoring out 1

𝑛+𝑝 in the equation in 14 above, complete the equation below.

1

𝑛 + 𝑝(_______________) = 𝑚

B C

A

𝑘

𝑝

𝑙

D

𝑚

𝑛



16. Simplify the equation in 15 above by using the appropriate substitution hence complete the

equation below by filling in the missing expression.

𝑚𝑝 + ____ = 𝑚2

Compare the equation in 16 above to the equation you wrote in question 12 above and hence, form a

statement of proportionality from the equations below:

17. 𝑘2 = 𝑚𝑝

18. 𝑙2 = 𝑚𝑛

19. Identify the side on ΔADB that corresponds to CA on ΔCAB

20. Given that all the triangles are right triangles, use the statements of proportionality in 17 and 18

above to state whether ΔABC, ΔDBA and ΔDAC are similar to each other or not. State the similarity

criterion you have used.



Answer keys Day 82:

1. 𝑟

2. 𝑟 + 𝑞

3. 𝑟 − 𝑞

4. 𝑝2 = 𝑟2 − 𝑞2

5. 𝑝2 = (𝑟 + 𝑞)(𝑟 − 𝑞)

6. 𝑝 =(𝑟+𝑞)(𝑟−𝑞)

𝑝

7. 𝑝

𝑟−𝑞=

𝑟+𝑞

𝑝

8. 𝐿𝑀̅̅ ̅̅

9. 𝑀𝐾̅̅ ̅̅ ̅

10. The two triangles are similar according to the S.A.S similarity criterion.

11. 𝑚 = 𝑛 + 𝑝

12. 𝑘2 + 𝑙2 = 𝑚2

13. 𝑝: 𝑛

14. 𝑚𝑝

𝑛+𝑝+

𝑚𝑛

𝑛+𝑝= 𝑚

15. 1

𝑛+𝑝(𝑚𝑝 + 𝑚𝑛) = 𝑚

16. 𝑚𝑝 + 𝑚𝑛 = 𝑚2

17. 𝑘

𝑝=

𝑚

𝑘

18. 𝑙

𝑛=

𝑚

𝑙

19. AD

20. All the triangles are similar to each other. S.A.S similarity criterion



In the figure below, O is the center of the circle, QS is the diameter, 𝑃𝑅 = 𝑘, 𝑅𝑂 = 𝑙 and 𝑂𝑃 = 𝑂𝑆 = 𝑚

Express the lengths of the following sides in terms of 𝑘, 𝑙 and 𝑚.

(a) RS̅̅̅̅

(b) OQ̅̅ ̅̅

(c) QR̅̅ ̅̅

S

P

Q R O

𝑚

𝑚 𝑙

𝑘



Answer keys

Day 82:

(a) 𝑙 + 𝑚

(b) 𝑚

(c) 𝑚 − 𝑙



1. Find the image of (18,-6) after dilation of a scale factor 3 about the origin.

2. Find the pre-image of (1,2) after a dilation of scale factor 1

5 about the origin.

3. A dilation has a scale factor of 6. Compare the size of the sides of the image.

4. Two similar triangles have a scale factor of 0.6. Find the hypotenuse of the smaller triangle if the

larger one’s is 5 in.

5. Two similar triangles are such that the ratio of the corresponding angles is 2.5. If the sides of one of

the triangle is 13 in, find all the interior angles of the other.



Answer Keys

Day 83:

12. (54, −18)

13. (5,10)

14. Image is 6 times larger than the pre-image

15. 3 in

16. All are equal to 60°



1. Identify a point at the left hand end and draw two lines intersecting an an angle less than 45°. Label

the point as N.

2. Draw a line to intersect the two lines from N. Let the line intersect the two are E and F respectively.

3. Draw another smaller line parallel to EF between N and EF. Label it as HT as shown.

4. Identify the image and the pre-image of this diagram shows a case of reduction of a line.

5. Identify the object and image distances.

6. Measure CT, CF, CH CE, HT and EF.

7. Find the following ratios

𝐶𝑇

𝐶𝐹,𝐶𝐸

𝐶𝐻,𝐸𝐹

𝐻𝑇

8. Compare the ratios in 7 above.

9. Is there a special name for the value in 7 above? Which one.

C

H

E

F

T



In this activity, students will draw two images under reduction and verify the scale factor under

reduction. They will work in groups of at least 4. Each group will require a protractor, a ruler, a pensil

and a plane paper.

Answer Keys

Day 83:

1 - 3. No response

4. Image; HT

Pre-image; EF

5. Object distance; CT and CF

Image distance; CT and CH

6. Different responses

7. The ratios should be approximately equal

8. They are approximately equal

9. Yes, Linear Scale factor




Identify if true or false

1. In reduction, the scale factor can be 2

3.

2. In reduction, the object can be between the image and the center of enlargement.

3. A negative scale factor implies reduction.

4. If the scale factor is −1

5, then center of reduction is between the image and the object.

5. In reduction, the object distance is more than the image distance.

Use the following information to answer questions 6 – 8.

A plan of rectangular shaped kitchen measuring 15 ft by 6 ft is to be drawn onto a paper. The length of

the kitchen on paper is 3 in.

6. Convert all the units of the kitchen to inches.

7. Find the linear scale factor relating the plan on paper and that on the ground.

8. Find the width of the kitchen on paper.



Use the following information to answer questions 9 -11.

The triangle 𝐴(3,0), 𝐵(6,9) and 𝐶(3,6) is the pre-image of 𝐴′𝐵′𝐶′ under the reduction the scale factor

of −1

3 about the origin.

9. Find the image of A.

10. Find the image of B.

11. Find the image of C.

12. Find the object distances, OA, OB, OC

13. Find the image distances 𝑂𝐴′, 𝑂𝐵′ and 𝑂𝐶′.

14. Find the ratios 𝑂𝐴′

𝑂𝐴,

𝑂𝐵′

𝑂𝐵 𝑎𝑛𝑑

𝑂𝐶′

𝑂𝐶 and compare them.



15. Draw the image and the pre-image on the same axes showing the center of reduction.



Use the following information to answer questions 16 - 20.

In the figure below, 𝐺𝐻 = 4 𝑖𝑛, 𝐹𝑅 = 16 𝑖𝑛. 𝐴𝐻 = 8 𝑖𝑛 and 𝐻𝑇 = 24 in. The diagram describes a

reduction.

16.Identify the image and the pre-image triangles.

17. Calculate the linear scale factor of the transformation.

18. Find the length of YT.

19. Find the length of AR

20. Find the length of AT.

A

F

H

G

R

T

Y



Answer keys

Day 83:

1. True

2. False

3. False

4. True

5. True

6. 180 in by 72 in

7. 1

60

8. 1.2 in

9. (−1,0)

10. (−2, −3)

11. (−1, −2)

12. 𝑂𝐴 = 3 𝑢𝑛𝑖𝑡𝑠, 𝑂𝐵 = 10.8 𝑢𝑛𝑖𝑡𝑠, 𝑂𝐶 = 6.7 𝑢𝑛𝑖𝑡𝑠

13. 𝑂𝐴′ = 1 𝑢𝑛𝑖𝑡 , 𝑂𝐵′ = 3.6 𝑢𝑛𝑖𝑡𝑠 and 𝑂𝐶′ = 2.2 𝑢𝑛𝑖𝑡𝑠.

14. 𝑂𝐴′

𝑂𝐴= 0.3333 ,

𝑂𝐵′

𝑂𝐵= 0.3333 𝑎𝑛𝑑

𝑂𝐶′

𝑂𝐶= 0.3283; They are approximately equal



15.

16. Image FHG

Pre-image RTY

17. 1

4

18. 16 in

19. 211

3 in

20. 32 in



Identify the location of the center of enlargement with respect to the image and the pre-image under

reduction of scale factor of 6

7.



Answer Keys

Day 83

The center of enlargement is on the left hands side of both the image and the pre-image




a) Find the value of 𝑥

b) Find the value of 𝑦

c) Find the value of z

2. Use the figure below to answer the questions that follow.

a) Given that ∆𝐴𝐸𝐷 has been dilated to form ∆𝐴𝐵𝐶, find the scale factor of dilation.

b) Find the value of 𝑥


20 𝑖𝑛 𝑥

𝑦 𝑧

12 𝑖𝑛 15 𝑖𝑛

A E B

D

C

10 𝑖𝑛 𝑥 𝑖𝑛



Answer Key Day 84:

1. a) 25 𝑖𝑛

b) 12 𝑖𝑛

c) 15 𝑖𝑛

2. a) 1.25

b) 2.5 𝑖𝑛



1. On the graph paper, draw the axes with a scale of one square representing one unit and label the

origin as O.

2. Plot a triangle with vertices at points 𝑆(−1, −1), 𝑇(2,1) and 𝑈(−1,2).

3. Multiply the coordinates of the vertices by -2 to get the coordinates of the image.

What are the coordinates of the image?

4. Mark the coordinates of the image and join them to form a triangle.

5. Using a ruler, measure the distance from the origin to each vertex of the image and the distance from

the origin to each vertex of the original triangle. Then fill the tables below

𝑂𝑆′ 𝑂𝑆 𝑂𝑇′ 𝑂𝑇 𝑂𝑈′ 𝑂𝑈

Length

𝑂𝑆′

𝑂𝑆

𝑂𝑇′

𝑂𝑇

𝑂𝑈′

𝑂𝑈



In this activity students will enlarge a triangle on a coordinate plane with scale factor of -2 with the

center of enlargement at the origin. Students will work in groups of at least three and each group is

required to have a graph paper, a ruler and a pencil.

Answer Keys Day 84:

1. No response

2.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

𝑺

𝑼

𝑻

O



3. 𝑆′(2,2), 𝑇(−4, −2) and 𝑈(2, −4).

4.

5.

𝑂𝑆′

𝑂𝑆

2

𝑂𝑇′

𝑂𝑇

2

𝑂𝑈′

𝑂𝑈

2

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

𝑺

𝑼

𝑻

O

𝑺′

𝑻′

𝑼′



1. A radius of a circle is 7 in. Find the diameter of its image after an enlargement with a linear scale

factor of 2.

Use the information below to answer questions 2-9

A contractor was given a map of a parking lot drawn on a piece of paper as shown below. This map was

to be transferred on a piece of land with side HG represented by 120 ft.

2. Find the linear scale factor that the contractor used to transfer the map.

3. Find the length of side AH on the piece of land in feet.

4. Find the length of side AB on the piece of land in feet.

5. Find the length of side BC on the piece of land in feet.

6. Find the length of side CD on the piece of land in feet.

3 𝑖𝑛 𝐴 5 𝑖𝑛 𝐵

C 𝐷

𝐸 F

𝐻 12 𝑖𝑛 𝐺

4 𝑖𝑛

8 𝑖𝑛

6 𝑖𝑛

2 𝑖𝑛 4.5 𝑖𝑛



7. Find the length of side DE on the piece of land in feet.

8. Find the length of side EF on the piece of land in feet.

9. Find the length of side FG on the piece of land in feet.

Use the diagram below to answer the questions 10-13

10. Find the center of enlargement.

11. Find the scale factor of enlargement leaving your answer in fraction form.

12. Find the length of side AC.

13. Find the length of side BC.

14. A rectangle measuring 4 𝑖𝑛 by 7 𝑖𝑛 is enlarged with a scale factor of 2. Find the measurements of the

image.

A

B C

E D

18 𝑖𝑛

2 𝑖𝑛

20 𝑖𝑛

27 𝑖𝑛



15. A rectangle measuring 4 𝑖𝑛 by 7 𝑖𝑛 is enlarged with a scale factor of -2. Find the measurements of

the image.


16. Find the scale factor of the enlargement given that ∆𝐽𝐾𝑁 is an enlargement of ∆𝐾𝐿𝑀

17. Find the center of enlargement.

19. Find the length of side KN.

20. Find the length of side JN.

2 𝑖𝑛


4 𝑖𝑛

𝐾

J

L

M

N



Answer Key Day 78:

1. 28 𝑖𝑛

2. 120

3. 60 𝑓𝑡

4. 50 𝑓𝑡

5. 45 𝑓𝑡

6. 40 𝑓𝑡

7. 20 𝑓𝑡

8. 30 𝑓𝑡

9. 80 𝑓𝑡

10. Vertex A

11. 10

9

12. 22.22 𝑖𝑛

13. 30 𝑖𝑛

14. 8 𝑖𝑛 by 14 𝑖𝑛

15. 8 𝑖𝑛 by 14 𝑖𝑛

16. −3

2

17. Point K

18. 6 𝑖𝑛

19. 4.5 𝑖𝑛

20. 4.5 𝑖𝑛



Parallelogram ABCD is an enlargement of parallelogram EGCH.

Find the length of EG

D 12 𝑖𝑛 𝐻 C

3 𝑖𝑛

9 𝑖𝑛

A B

E G



Answer Keys

Day 84:

4 𝑖𝑛

248


Geometry


Week 17


249

Week 17

Weekly Assessments

250

Week #17

1. Consider the figure below, EC and AB are parallel and AB is twice EC.

a. Show that corresponding angles of triangles ECD and ABD are equal.

b. Show that the two angles are similar

c. Find the scale factor relating the sides of the two angles.

2. a).What is reduction?

b). Compare the angles of two triangles that are related by reduction c). Is reduction a rigid transformation? Explain.

3. A model of a house measures 2.8 in by 1.2 in. If

the real house has a length of 1100in

a). Find the actual measurement of its width b). Find the actual area of the house.

A B

D

C E

251

4. In the figure below, 𝑆𝑄 = 18𝑖𝑛, 𝑆𝑇 = 9 𝑖𝑛 and

𝑅𝑄 = 6 𝑖𝑛.

a). Find the size of 𝑆𝑅

b), Find the size of SP

5. Use the figure below to determine ratio

between corresponding sizes.

a). Identify sides of triangle ACD corresponding to AB and CB of triangle ABC. d) Determine the ratios between the sides in a above given that ACB is an image of ACD. in terms of 𝑎, 𝑏, 𝑐, 𝑐, 𝑑 𝑜𝑟 𝑒where necessary.

6. Use similarity and congruence to show that the diagonal of a square divides it.

Q P

R

S

T

a A B

C

D

b

c d

252

Week 17 - Keys

Weekly Assessments

253

Week #17 KEY

1. Consider the figure below, EC and AB are parallel, and AB is twice EC

a. Show that corresponding angles of triangles ECD and ABD are equal.

Angle D is common ∠𝐴 = ∠𝐶𝐸𝐷 (corresponding angles since EC and AB are parallel) ∠𝐵 = ∠𝐸𝐶𝐷 (corresponding angles since EC and AB are parallel)

b. Show that the two angles are similar

Corresponding angles are equal and ACD is dilated to get ABD. Further, there is zero translation hence a dilation and a translation implies similarity

c. Find the scale factor relating the sides of the two angles.

Scale factor = 𝐴𝐵

𝐸𝐶=

2

1= 2

2. a).What is reduction

A transformation where the sides of one figure a reduced a scale factor of less than absolute 1 to get an image b). Compare the angles of two triangles that are related by reduction They are the same c). Is reduction a rigid transformation? Explain. No, the image and the pre-image of not of the same size

3. A model of a house measures 2.8 in by 1.2 in. If

the real house has a length of 1540in

a). Find the actual measurement of its width 660 in b). Find the actual area of the house. 1016400 sq. in

A B

D

C E

254

4. In the figure below, 𝑆𝑄 = 18𝑖𝑛, 𝑆𝑇 = 9 𝑖𝑛 and

𝑅𝑄 = 6 𝑖𝑛.

a). Find the size of 𝑆𝑅

12 in

b), Find the size of SP 13.5 in

5. Use the figure below to determine ratio

between corresponding sizes.

a). Identify sides of triangle ACD corresponding to AB and CB of triangle ABC.

AB corresponds to AD CB corresponds to AC

d) Determine the ratios between the sides in a above given that ACB is an image of ACD in terms of 𝑎, 𝑏, 𝑐, 𝑐, 𝑑 𝑜𝑟 𝑒where necessary. 𝐴𝐵

𝐴𝐷=

𝑎

𝑑;

𝐶𝐵

𝐴𝐶=

𝑐−𝑑

𝑒

6. Use similarity and congruence to show that the diagonal of a square divides it.

The diagonal is common to the two triangles The other corresponding sides of the triangle equal since they are part of the sides of the square which are always equal by the definition of the square. Thus, we get a postulate SSS This implies that the two triangles on either side of the diagonal are equal

Q P

R

S

T

a A B

C

D

b

c d

e

Unit 5 Test Name ____________________________________


Questions:

1. Construct a perpendicular bisector to the given line.

2. Identify two sides that are congruent in the triangle.

3. The point Q is a bisector of the line PR. If PQ=11cm, what is the length of QR?

4. Define midpoint!

R

P Q

P Q

R

Unit 5 Test Name ____________________________________


5. ∆𝐴𝐵𝐶 is dilated to form ∆𝐶𝐷𝐸. Find the scale factor of dilation.

6. AB is parallel to ST. 𝑆𝑈 = 8𝑖𝑛, ST= 6𝑖𝑛, and AB= 3𝑖𝑛. What is the length of AS?

Unit 5 Test Name ____________________________________


7. R and E are the midpoints of YP and YH respectively. Find the size of angle REH.

8. Find the length of 𝐿′𝑀′.

Unit 5 Test Name ____________________________________


9. The point (4,10) is dilated by a scale factor of 1.5.

Find the image point!

10. Identify the corresponding sides.

11. Are the two figures similar? If yes, find the scale factor!

12. An elephant is 14 feet tall and its shadow is 7 feet long at exactly 10.30 am. John’s shadow is 4.5

feet at the same time. Find the John’s height.

Unit 5 Test Name ____________________________________


13. When a building casts a shadow 80 feet long, a man 6 feet tall casts a shadow of length 8 feet.

How tall is the building?

14. A dilation has a scale factor of 3.15. Compare the size of the image and that of the pre-image.

15. Identify two triangles.

16. Find the length of KL.

A B C

D E

Unit 5 Test Name ____________________________________


17. ΔABC is a right triangle and BD⊥AC. Find the measure of ∠BAD.

18. Define congruent triangles.


𝑅𝐶!

Unit 5 Test Name ____________________________________


20. Parallelogram ABCD is an enlargement of parallelogram EGCH. Find the length of EG.

Unit 5 Test Name ____________________________________


Answers:

1.

2. PQ and PR

3. QR=11cm

4. Midpoint is a point on a line segment that divides it into two equal parts. The point is equidistant

from both endpoints of the segment.

5. 1

2

6. 4in

7. 137°

8. 6 units

9. (6, 15)

10. XE and RQ

EW and QT

WX and TR

11. Yes! The scale factor is 1.5.

12. 9 feet

13. 60 feet

14. The image is larger than the pre-image by a linear scale factor factor of 3.15.

15. △ 𝐴𝐶𝐸 and △ 𝐵𝐶𝐷

16. KL=20in

17. ∠BAD=47°

18. Two triangles are congruent when they have the same size and shape.

19. 1

5

20. 4 in

Geometry Complete Unit 5 - High School Math Teachers...Unit 5 Pacing Chart...

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