Goal Your Notes VOCABULARY -...

Ratios, Proportions, and the Geometric MeanGoal p Solve problems by writing and solving proportions.

VOCABULARY

Ratio of a to b If a and b are two numbers or quantities and b Þ 0, then the ratio of a to b is a }

b .

Proportion An equation that states that two ratios are equal is a proportion.

Means, extremes In the proportion a } b 5 c }

d , b and c

are the means, and a and d are the extremes.

Geometric mean The geometric mean of two positive numbers a and b is the positive number

x that satisfies a } x 5 x }

b .

Simplify the ratio. (See Table of Measures, p. 921)

a. 76 cm : 8 cm b. 4 ft } 24 in.

Solution

a. Write 76 cm : 8 cm as 76 cm

8 cm. Then divide out the

units and simplify.

76 cm

8 cm 5

19

2 5 19 : 2

b. To simplify a ratio with unlike units, multiply by a conversion factor.

4 ft } 24 in. 5 4 ft

} 24 in. p 12 in.

1 ft 5

48

24 5

2

1

Example 1 Simplify ratios

For help with conversion factors, see p. 992.

144 Lesson 6.1 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

6.1

Your Notes

LAH_GE_11_FL_NTG_Ch06_144-175.indd 144 12/16/09 12:09:31 AM

Painting You are painting barn doors. You know that the perimeter of the doors is 64 feet and that the ratio of the length to the height is 3 : 5. Find the area of the doors.

SolutionStep 1 Write expressions for the length and height.

Because the ratio of the length to height is 3 : 5, you can represent the length by 3 x and the height by 5 x.

Step 2 Solve an equation to find x.

2l 1 2w 5 P Formula for perimeter

2( 3 x) 1 2( 5 x) 5 64 Substitute.

16 x 5 64 Multiply and combine like terms.

x 5 4 Divide each side by 16 .

Step 3 Evaluate the expressions for the length and height. Substitute the value of x into each expression.

Length: 3 x 5 3 ( 4 ) 5 12

Height: 5 x 5 5 ( 4 ) 5 20

The doors are 12 feet long and 20 feet high, so the area is 12 p 20 5 240 ft2 .

Example 2 Use a ratio to find a dimension

1. 4 meters to 18 meters 2. 33 yd : 9 ft

2 to 9 11 : 1

3. The perimeter of a rectangular table is 21 feet and the ratio of its length to its width is 5 : 2. Find the length and width of the table.

length: 7.5 feet, width: 3 feet

Checkpoint In Exercises 1 and 2, simplify the ratio.

Copyright © Holt McDougal. All rights reserved. Lesson 6.1 • Geometry Notetaking Guide 145

Your Notes

LAH_GE_11_FL_NTG_Ch06_144-175.indd 145 3/3/09 8:29:24 PM

The measures of the angles in nBCD are in the extended ratio of 2 :3 :4. Find the measures of the angles.

Solution

2x 8

3x 8 4x 8

Begin by sketching the triangle. Then use theextended ratio of 2 : 3 : 4 to label the measures as 2 x8, 3 x8, and 4 x8.

2 x8 1 3 x8 1 4 x8 5 1808 Triangle Sum Theorem

9 x 5 180 Combine like terms.

x 5 20 Divide each side by 9 .

The angle measures are 2( 208 ) 5 408 , 3( 208 ) 5 608 , and 4( 208 ) 5 808 .

Example 3 Use extended ratios

4. A triangle’s angle measures are in the extended ratio of 1 : 4 : 5. Find the measures of the angles.

188, 728, 908

Checkpoint Complete the following exercise.

A PROPERTY OF PROPORTIONS

1. Cross Products Property In a proportion, the product of the extremes equals the product of the means.

If a } b 5 c } d where b Þ 0 and d Þ 0, then ad 5 bc .

2 } 3 5 4 } 6 3 p 4 5 12

2 p 6 5 12


Your Notes


Solve the proportion.

a. 3 } 4 5 x } 16 Original proportion

3 p 16 5 4 p x Cross Products Property

48 5 4 x Multiply.

12 5 x Divide each side by 4 .

b. 3 } x 1 1 5 2 } x Original proportion

3 p x 5 2 (x 1 1) Cross Products Property

3 x 5 2 x 1 2 Distributive Property

x 5 2 Subtract 2x from each side.

Example 4 Solve proportions

Bowling You want to find the total number of rows of boards that make up 24 lanes at a bowling alley. You know that there are 117 rows in 3 lanes. Find the total number of rows of boards that make up the 24 lanes.

SolutionWrite and solve a proportion involving two ratios that compare the number of rows with the number of lanes.

117 } 3 5 n } 24 number of rows

number of lanes Write proportion.

117 p 24 5 3 p n Cross Products Property

936 5 n Simplify.

There are 936 rows of boards that make up the 24 lanes.

Example 5 Solve a real-world problem

GEOMETRIC MEAN

The geometric mean of two positive numbers a and b is

the positive number x that satisfies a } x 5 x } b .

So, x2 5 ab and x 5 Ï}

ab .

In part (a), you could multiply each side by the denominator, 16. Then 16 p 3 } 4 5 16 p x } 16

so 12 5 x.


Your Notes


Find the geometric mean of 16 and 48.

Solutionx 5 Ï

}

ab Definition of geometric mean

5 Ï}

16 p 48 Substitute 16 for a and 48 for b.

5 Ï}

16 p 16 p 3 Factor.

5 16 Ï}

3 Simplify.

The geometric mean of 16 and 48 is 16 Ï}

3 ø 27.7 .

Example 6 Find a geometric mean

5. Solve 8 } y 5 2 } 5 . 6. Solve x 2 3 } 3 5 2x

} 9 .

y 5 20 x 5 9

7. A small gymnasium contains 10 sets of bleachers. You count 192 spectators in 3 sets of bleachers and the spectators seem to be evenly distributed. Estimate the total number of spectators.

about 640 spectators

8. Find the geometric mean of 14 and 16.

4 Ï}

14 ø 15.0

Checkpoint Complete the following exercises.


Homework

Your Notes


6.2 Use Proportions to Solve Geometry ProblemsGoal p Use proportions to solve geometry problems.

ADDITIONAL PROPERTIES OF PROPORTIONS

2. Reciprocal Property If two ratios are equal, then their reciprocals are also equal.

If a } b 5 c } d , then b } a 5 d } c .

3. If you interchange the means of a proportion, then you form another true proportion.

If a } b 5 c } d , then a } c 5 b } d .

4. In a proportion, if you add the value of each ratio’s denominator to its numerator, then you form another true proportion.

If a } b 5 c } d , then a 1 b } b 5 c 1 d

} d .


VOCABULARY

Scale drawing A scale drawing is a drawing that is the same shape as the object it represents.

Scale The scale is a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object.

Your Notes


1. In Example 1, find 2. In Example 2,

KL } GH 5 JK } JG . Find GH.the value of x.

x 5 13.5 GH 5 16


In the diagram, AC }

DF 5 BC

} EF

. Write

A

C

BD

F

E

x

12

189four true proportions.

Because AC } DF 5 BC

} EF , then 12 } 18 5 9 } x .

Reciprocal Property: The reciprocals are equal, so 18 } 12 5 x } 9 .

Property 3: You can interchange the means, so 12 } 9 5 18

} x .

Property 4: You can add the denominators to the

numerators, so 30 } 18

5 9 1 x } x .

Example 1 Use properties of proportions

In the diagram, JL } LH

5 JK } KG

.

Find JH and JL.

HG

KL

J

15

125

2

x

JL } LH

5 JK } KG Given

JL 1 LH } LH 5 JK 1 KG }

KG Property of Proportions

(Property 4)

x } 2 5 15 1 5 } 5 Substitution Property of Equality

5x 5 2(15 1 5) Cross Products Property

x 5 8 Solve for x.

So JH 5 8 and JL 5 8 2 2 5 6 .

Example 2 Use proportions with geometric figures


Your Notes



Keys The length of the key in the scale drawing is 7 centimeters. The length of the actual key is 4 centimeters. What is the scale of the drawing?

SolutionTo find the scale, write the ratio of a length in the drawing to an actual length , then rewrite the ratio so that the denominator is 1.

length in drawing

length of key 5 7 cm

} 4 cm 5 7 4 4 } 4 4 4 5 1.75

} 1

The scale of the drawing is 1.75 cm : 1 cm .

Example 3 Find the scale of a drawing

Maps The scale of the map at

Cooley

WestbrookJackson

Greenbow

the right is 1 inch : 8 miles. Find the actual distance from Westbrook to Cooley.

SolutionUse a ruler. The distance from Westbrook to Cooley on the map is about 1.25 inches . Let x be the actual distance in miles.

1.25 in. } x mi 5 1 in.

} 8 mi distance on map

actual distance

x 5 1.25(8) Cross Products Property

x 5 10 Simplify.

The actual distance from Westbrook to Cooley is about 10 miles .

Example 4 Use a scale drawing

3. In Example 3, suppose the length of the key in the scale drawing is 6 centimeters. Find the new scale of the drawing.

1.5 cm : 1 cm


Your Notes


4. Two landmarks are 130 miles from each other. The landmarks are 6.5 inches apart on a map. Find the scale of the map.

1 inch : 20 miles

5. Your friend has a model of the Sunsphere that is 5 inches tall. What is the approximate diameter of the dome on your friend’s model?

about 1.4 inches


Scale Model You buy a 3-D scale model of the Sunsphere in Knoxville, TN. The actual building is 266 feet tall. Your model is 20 inches tall, and the diameter of the dome on your scale model is about 5.6 inches.

a. What is the diameter of the actual dome?

b. How many times as tall as your model is the actual building?

Solution

a. 20 in. } 266 ft 5 5.6 in. } x ft measurement on model

measurement on actual building

20 x 5 1489.6 Cross Products Property

x ø 74.5 Divide each side by 20 .

The diameter of the actual dome is about 74.5 feet.

b. To simplify a ratio with unlike units, multiply by a conversion factor.

266 ft } 20 in. 5 266 ft

} 20 in. p 12 in.

} 1 ft

5 159.6

The actual building is 159.6 times as tall as the model.

Example 5 Solve a multi-step problem


Homework

Your Notes



In the diagram, nABC , nDEF.

A

B

EC

D F

10 8

12

15 12

18

a. List all pairs of congruent angles.

b. Check that the ratios of corresponding side lengths are equal.

c. Write the ratios of the corresponding side lengths in a statement of proportionality.

Solution

a. ∠A > ∠ D , ∠B > ∠ E , ∠C > ∠ F

b. AB } DE 5 10

} 15 5 2 } 3 BC } EF 5 8 } 12 5 2 } 3

CA } FD 5 12

} 18 5 2 } 3

c. The ratios in part (b) are equal, so

AB } DE 5 BC

} EF 5 CA } FD .

Example 1 Use similarity statements

In a statement of proportionality, any pair of ratios forms a true proportion.

1. Given nPQR , nXYZ, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality.

∠P > ∠X, ∠Q > ∠Y, ∠R > ∠Z; PQ } XY

5 QR } YZ

5 RP } ZX


Goal p Use proportions to identify similar polygons.

Use Similar Polygons

VOCABULARY

Similar polygons Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional.

Scale factor of two similar polygons If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor.

6.3

Your Notes


In the diagram, nBCD , nRST. Find the value of x.

Solution

C

B D

S

R T

12

24

13

x

105

The triangles are similar, so the corresponding side lengths are proportional .

BC

RS 5

STCD

Write proportion.

12

24 5

x13

Substitute.

12x 5 312 Cross Products Property

x 5 26 Solve for x.

Example 3 Use similar polygons

Determine whether the polygons A B

M

C

D

J K

L

218

12

8

8

14

14

14

are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKLM.

SolutionStep 1 Identify pairs of congruent angles.

From the diagram, you can see that ∠B > ∠ K , ∠C > ∠ L , and ∠D > ∠ M . Angles A and J are right angles, so ∠ A > ∠ J . So, the corresponding angles are congruent .

Step 2 Show that corresponding side lengths are proportional.

AB } JK 5 8 } 14 5 4 } 7 BC

} KL 5 8 } 14 5 4 } 7

AD } JM 5 12

} 21 5 4 } 7

CD } LM 5 8 } 14 5 4 } 7

The ratios are equal, so the corresponding side lengths are proportional .

So ABCD , JKLM . The scale factor of ABCD to JKLM

is 4 } 7 .

Example 2 Find the scale factor

There are several ways to write the proportion. For example, you could

write BD } RT 5 CD

} ST .


Your Notes



2. What is the scale factor of LMNP to FGHJ?

4 } 5 F

GM

NP

L x

HJ

15 12

16

12

20

40

15

3. Find the value of x.

32

Checkpoint In the diagram, FGHJ , LMNP.

THEOREM 6.1: PERIMETERS OF SIMILAR POLYGONS

If two polygons are similar, then the ratio of their perimeters is equal to the

K L

MN

PQ

RS

ratios of their corresponding side lengths.

If KLMN , PQRS, then

KL 1 LM 1 MN 1 NK }} PQ 1 QR 1 RS 1 SP 5 KL }

PQ 5 LM

} QR

5 MN }

RS 5 NK

} SP

.

Basketball A larger cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wide and 25 feet long. The old court was similar in shape, but only 16 feet wide.

a. Find the scale factor of the new court to the old court.

b. Find the perimeters of the new court and the old court.

Solutiona. Because the new court will be similar to the old court,

the scale factor is the ratio of the widths, 20 } 16 5 5 } 4 .

b. The new court’s perimeter is 2(20) 1 2(25) 5 90 feet. Use Theorem 6.1 to find the perimeter x of the old court.

90 } x 5 5 } 4 Use Theorem 6.1 to write a proportion.

x 5 72 Simplify.

The perimeter of the old court was 72 feet.

Example 4 Find perimeters of similar figures

Your Notes


CORRESPONDING LENGTHS IN SIMILAR POLYGONS

If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

In the diagram, nFGH , nJGK.

HLF

J KM

G

55

14

88

Find the length of the altitude } GL .

SolutionFirst, find the scale factor of nFGH to nJGK.

FH

JK 5 8 1 8

} 5 1 5 5 16 } 10 5 8 } 5

Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion.

GL } GM 5 8 } 5 Write proportion.

GL

14 5 8 } 5 Substitute 14 for GM.

GL 5 22.4 Multiply each side by 14 and simplify.

The length of altitude } GL is 22.4 .

Example 5 Use a scale factor

4. Find the perimeter of nWXY.

PWY

X

R

Q

4072

63

81

The perimeter of nWXY is 120.

5. Find the length of median } QS .

W

P

RS

Q

YZ

X

39

24

16

QS 5 26

Checkpoint In the diagrams, nPQR , nWXY.


Homework

Your Notes


Find the eighth and ninth terms of the Fibonacci sequence.

SolutionEach term of the Fibonacci sequence after the second term is the sum of the previoue two terms.

3rd term 1 1 1 5 2 Sum of 1st and 2nd terms

4th term 1 1 2 5 3 Sum of 2nd and 3rd terms

5th term 2 1 3 5 5 Sum of 3rd and 4th terms

6th term 3 1 5 5 8 Sum of 4th and 5th terms

7th term 5 1 8 � 13 Sum of 5th and 6th terms

8th term 8 � 13 � 21 Sum of 6th and 7th terms

9th term 13 � 21 � 34 Sum of 7th and 8th terms

Example 1 Find terms in the Fibonacci sequence

1. What are the tenth and eleventh terms of the Fibonacci sequence?

55; 89

Checkpoint Find terms of the Fibonacci sequence.

VOCABULARY

Fibonacci sequence The sequence 1, 1, 2, 3, 5, ..., which uses the rule that each term after the second term is the sum of the previous two terms

Golden ratio The irrational number 1.618... approached by the ratios of consecutive terms from the Fibonacci sequence.

Golden rectangle A rectangle whose sides form a golden ratio.

Goal p Use the Fibonacci sequence and the golden ratio.

Fibonacci Sequence and the Golden Ratio

Copyright © Holt McDougal. All rights reserved. 6.3 Foucs on Patterns • Geometry Notetaking Guide 157

Your Notes

Focus On PatternsUse after Lesson 6.3


Find the ratios of consecutive terms in the Fibonacci sequence using the seventh, eighth, and ninth terms. Round to the nearest 0.001.

SolutionThe seventh, eighth, and ninth terms of the Fibonacci sequence are 13 , 21 , and 34 .

8th term } 7th term

5 21 }

13 5 1.615 }

9th term }

8th term 5

34 }

21 5 1.619 }

Example 2 Find ratios of terms in the Fibonacci sequence

2. Which two consecutive terms of the Fibonacci sequence will give you a ratio of approximately 1.618?

ninth and tenth terms


Show that the figure is nearly a golden rectangle.

SolutionFor a rectangle to be a golden rectangle,

length

} width

5 width 1 length

}} length

≈ 1.618 } .

For the figure shown,

6.5 } 4

5 1.625 , and 6.5 1 4

} 6.5

≈ 1.615 } .

Yes, the figure is nearly a golden rectangle.

Example 3 The golden rectangle

3. Is a 15 m 3 13 m rectangle nearly a golden rectangle?

No


6.5 feet

4feet

158 6.3 Focus on Patterns • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Homework

Your Notes


6.4 Prove Triangles Similar by AA


POSTULATE 22: ANGLE-ANGLE (AA) SIMILARITY POSTULATE K L

J X

Y ZIf two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. nJKL , nXYZ

Determine whether the triangles

B C

AE D

F

688

228are similar. If they are, write a similarity statement. Explain your reasoning.

Solution

Because they are both right angles, ∠ B and ∠ E are congruent.

By the Triangle Sum Theorem, 688 1 908 1 m∠A 5 1808, so m∠A 5 228 . Therefore, ∠A and ∠ D are congruent.

So, nABC , nDEF by the AA Similarity Postulate .

Example 1 Use the AA Similarity Postulate

Goal p Use the AA Similarity Postulate.

Checkpoint Determine whether the triangles are similar. If they are, write a similarity statement.

1. PG

F H

R

Q

718 788

318

788

2.

GP

R

F

H

Q

2781108

278

538

nFGH , nRQP not similar

Your Notes


Show that the two triangles are similar.

a. nRTV and nRQS b. nLMN and nNOP

S

R

T V

Q498

498

M

O

PNL

Solutiona. You may find it helpful to redraw the triangles

separately.

Because m∠ RTV and m∠ Q both equal 498, ∠ RTV > ∠ Q . By the Reflexive Property, ∠R > ∠ R .

So, nRTV , nRQS by the AA Similarity Postulate .

b. The diagram shows ∠L > ∠ ONP . It also shows that } MN i

} OP so ∠ LNM > ∠ P by the Corresponding

Angles Postulate.

So, nLMN , nNOP by the AA Similarity Postulate .

Example 2 Show that triangles are similar

3. Show that nBCD , nEFD.

E

F

B

C

D

Because they are both right angles, ∠C > ∠F.

You know that ∠CDB > ∠FDE by the Vertical Angles Congruence Theorem.

So, nBCD , nEFD by the AA Similarity Postulate.



Your Notes



4. In Example 3, how long is the shadow of a person that is 4 feet 9 inches tall?

3 feet


Height A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair?

SolutionThe lifeguard and the chair form sides of two right triangles with the ground, as shown below. The sun’s rays hit the lifeguard and the chair at the same angle. You have two pairs of congruent angles , so the triangles are similar by the AA Similarity Postulate .

48 in. 6 ft

x ft

6 ft 4 in.

You can use a proportion to find the height x. Write 6 feet 4 inches as 76 inches so you can form two ratios of feet to inches.

x ft

76 in. 5

6 ft

48 in. Write proportion of side lengths.

48 x 5 456 Cross Products Property

x 5 9.5 Solve for x.

The chair is 9.5 feet tall.

Example 3 Using similar triangles

Homework

Your Notes


Is either nDEF or nGHJ similar to nABC?

H

J GA

B

C

D F

E12

8 918

24

1684 6

SolutionCompare nABC and nDEF by finding ratios of corresponding side lengths.

Shortest sides Longest sides Remaining sides

AB } DE 5 8 } 4 5 2 CA

} FD 5 12 } 8 5 3 } 2 BC

} EF 5 9 } 6 5 3 } 2

The ratios are not all equal , so nABC and nDEF are not similar .

Compare nABC and nGHJ by finding ratios of corresponding side lengths.

Shortest sides Longest sides Remaining sides

AB } GH 5 8 } 16 5 1 } 2 CA

} JG 5 12 } 24 5 1 } 2 BC

} HJ 5 9 } 18 5 1 } 2

All the ratios are equal , so n ABC , n GHJ .

Example 1 Use the SSS Similarity Theorem

When using the SSS Similarity Theorem, compare the shortest sides, the longest sides, and then the remaining sides.

6.5 Prove Triangles Similar by SSS and SASGoal p Use the SSS and SAS Similarity Theorems.

THEOREM 6.2: SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM

B C

A

R

S T

If the corresponding side lengths of two triangles are proportional , then the triangles are similar.

If AB } RS 5 BC

} ST 5 CA } TR , then nABC , nRST.


Your Notes



Find the value of x that

A

B

E

C D F

x 2 3

2x 1 310

104

20makes nABC , nDEF.

SolutionStep 1 Find the value of x that makes corresponding side

lengths proportional.

4

10 5 x 2 3

20 Write proportion.

4 p 20 5 10 (x 2 3) Cross Products Property

80 5 10 x 2 30 Simplify.

11 5 x Solve for x.

Step 2 Check that the side lengths are proportional when x 5 11 .

AB 5 x 2 3 5 8 DF 5 2x 1 3 5 25

BC } EF 0 AB

} DE 4 } 10 5 8 } 20 ✓ BC } EF 0 AC

} DF 4 } 10 5 10 } 25 ✓

When x 5 11 , the triangles are similar by the SSS Similarity Theorem .

Example 2 Use the SSS Similarity Theorem

1. Which of the three triangles

A

P

Q

R

Y

X Z

B

C

45

35

30

18 21

27

42

54

38

are similar?

nPQR , nZXY

2. Suppose AB is not given in nABC. What value of AB would make nABC similar to nQRP?

AB 5 36


Your Notes


Birdfeeder You are drawing a design

32 cm

20 cm20 cm

32 cm

878

878

A C

B

E

D F

for a birdfeeder. Can you construct the top so it is similar to the bottom using the angle measure and lengths shown?

Solution

Both m∠ B and m∠ E equal 878, so ∠ B > ∠ E . Next, compare the ratios of the lengths of the sides that include ∠B and ∠E.

AB

DE 5

EF

BC 5 32

} 20 5 8 } 5

The lengths of the sides that include ∠B and ∠E are proportional .

So, by the SAS Similarity Theorem , nABC , nDEF.Yes, you can make the top similar to the bottom.

Example 3 Use the SAS Similarity Theorem

THEOREM 6.3: SIDE-ANGLE-SIDE (SAS) SIMILARITY THEOREM

If an angle of one triangle is X

YZ

M

NP

congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional , then the triangles are similar.

If ∠X > ∠M, and ZX } PM 5 XY

} MN , then nXYZ , nMNP.

3. In Example 3, suppose you use equilateral triangles on the top and bottom. Are the top and bottom similar? Explain.

Yes, the top and bottom are similar. If the side length of the top is a and the side length of the bottom is b, the ratios of the side lengths are

a } b and the angles are all 608. The triangles are

similar by SAS or SSS.



Your Notes



TRIANGLE SIMILARITY POSTULATE AND THEOREMS

AA Similarity Postulate If ∠A > ∠D and ∠B > ∠E, then nABC , nDEF.

SSS Similarity Theorem If AB } DE 5 BC

} EF 5 AC } DF ,

then nABC , nDEF.

SAS Similarity Theorem If ∠A > ∠D and

AB } DE 5 AC

} DF , then nABC , nDEF.

Tell what method you would use to P

RS

T

Q9

12

18

24

show that the triangles are similar.

SolutionFind the ratios of the lengths of the corresponding sides.

Shorter sides QR } RS

5 9 } 12

5 3 } 4

Longer sides PR } RT

5 18 } 24

5 3 } 4

The corresponding side lengths are proportional . The included angles ∠PRQ and ∠TRS are congruent because they are vertical angles. So, nPQR , nTSR by the SAS Similarity Theorem .

Example 4 Choose a method

4. Explain how to show nJKL , nLKM.

J MK

L

49

5628

32

16 Show that the corresponding side

lengths are proportional, then use the SSS Similarity Theorem to show nJKL , nLKM.


To identify corresponding parts, redraw the triangles so that the corresponding parts have the same orientation.

P

R

S

R

T

Q9 12

1824

Homework

Your Notes


In the diagram, } QS i } UT , RQ 5 10, RS 5 12, and ST 5 6. What is the length of } QU ?

U

Q S

T

R

10 12

6

Solution

RQ

} QU

5 RS } ST

Triangle Proportionality Theorem

QU10

5 12

6 Substitute.

60 5 12 p QU Cross Products Property

5 5 QU Divide each side by 12 .

Example 1 Find the length of a segment

6.6 Use Proportionality TheoremsGoal p Use proportions with a triangle or parallel lines.

THEOREM 6.4: TRIANGLE PROPORTIONALITY THEOREM

If a line parallel to one side of a

S U

R

TQ

triangle intersects the other two sides, then it divides the two sides proportionally . If } TU i } QS , then RT

} TQ

5 RU }

US .

THEOREM 6.5: CONVERSE OF THE TRIANGLE PROPORTIONALITY THEOREM

If a line divides two sides of a

S U

R

TQ

triangle proportionally, then it is parallel to the third side .

If RT } TQ 5 RU

} US , then } TU i } QS .


Your Notes



1. Find the length of } KL .

N

L

J

K

M

18

24

44

KL 5 33

2. Determine whether } QT i } RS . 18

36

80

38

PQ

R

T

S

No; }

QT is not parallel to }

RS .


Aerodynamics A spoiler for a remote

Not drawn to scale

A

BC

E

D

controlled car is shown where AB 5 31 mm, BC 5 19 mm, CD 5 27 mm, and DE 5 23 mm. Explain why } BD is not parallel to } AE .

SolutionFind and simplify the ratios of lengths determined by } BD .

CD } DE 5 27

} 23 CB } BA 5 31

} 19

Because 27 } 23 Þ 31 } 19 , } BD } AE .

Example 2 Solve a real-world problem

Þ

Your Notes


Farming A farmer’s land is A

B

CWest Border

North Border

Interstate

South BorderD

E

F

2000 2500

3000

divided by a newly constructed interstate. The distances shown are in meters. Find the distance CA between the north border and the south border of the farmer’s land.

Use Theorem 6.6.

CB } BA 5 DE

} EF Parallel lines divide transversals proportionally.

BA

CB 1 BA 5

EF

DE 1 EF Property of proportions

(Property 4)

CA

2000 5 3000 1 2500

}} 2500 Substitute.

CA

2000 5 5500

} 2500 Simplify.

CA 5 4400 Multiply each side by 2000 and simplify.

The distance between the north border and the south border is 4400 meters.

Example 3 Use Theorem 6.6

THEOREM 6.6 U

V X Z

W Yr s t

l

m

If three parallel lines intersect two transversals, then they divide the transversals proportionally .

UW } WY 5 VX

} XZ

THEOREM 6.7 A

C B

DIf a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. AD

} DB 5 CA }

CB


Your Notes



In the diagram, ∠DEG > ∠GEF.

D FGx

8 12

E

14

Use the given side lengths to find the length of } DG .

SolutionBecause ###$ EG is an angle bisector of ∠DEF, you can apply Theorem 6.7. Let GD 5 x. Then GF 5 14 2 x .

GF } GD 5 EF

} ED Angle bisector divides opposite side proportionally.

x

14 2 x 5 12

} 8 Substitute.

12 x 5 112 2 8 x Cross Products Property

x 5 5.6 Solve for x.

Example 4 Use Theorem 6.7

3.

A

B

24

27

32

AB 5 36

4.

A

B

C

D

1

1

2

AB 5 Ï}

2

Checkpoint Find the length of } AB .

Homework

Your Notes


Similarity Transformations and Coordinate Geometry

6.7Goal p Perform dilations.

COORDINATE NOTATION FOR A DILATION

You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor.

If 0 < k < 1, the dilation is a reduction . If k > 1, the dilation is an enlargement .

VOCABULARY

Dilation A dilation is a transformation that stretches or shrinks a figure to create a similar figure.

Center of dilation In a dilation, a figure is enlarged or reduced with respect to a fixed point called the center of dilation.

Scale factor of a dilation The scale factor k of a dilation is the ratio of a side length of the image to the corresponding side length of the original figure.

Reduction A dilation where 0 < k < 1 is a reduction.

Enlargement A dilation where k > 1 is an enlargement.


Your Notes


Draw a dilation of quadrilateral ABCD with vertices A(2, 0), B(6, 24), C(8, 2), and D(6, 4). Use a scale factor of 1 }

2 .

First draw ABCD. Find the dilation of each vertex by

multiplying its coordinates by 1 } 2 . Then draw the dilation.

(x, y) → 1 1 } 2 x, 1 } 2 y 2

x

y

1

1L

P

A

D

C

B

N

M

A(2, 0) → L (1, 0) B(6, 24) → M (3, 22) C(8, 2) → N (4, 1) D(6, 4) → P (3, 2)

Example 1 Draw a dilation with a scale factor greater than 1

All of the dilations in this lesson are in the coordinate plane and each center of dilation is the origin.

A triangle has the vertices A(2, 21), B(4, 21), and C(4, 2). The image of nABC after a dilation with a scale factor of 2 is nDEF.

a. Sketch nABC and nDEF.b. Verify that nABC and nDEF are similar.

Solution

x

y

1

1

EA

C

B

F

D

a. The scale factor is greater than 1, so the dilation is an enlargement .

(x, y) → ( 2 x, 2 y)

A(2, 21) → D(4, 22) B(4, 21) → E(8, 22) C(4, 2) → F(8, 4)

b. Because ∠ B and ∠ E are both right angles, ∠ B > ∠ E . Show that the lengths of the sides that include ∠ B and ∠ E are proportional.

AB

DE 0 BC

EF 2

4 5 3

6 ✓

The lengths are proportional. So, nABC , nDEF by the SAS Similarity Theorem .

Example 2 Verify that a figure is similar to its dilation


Your Notes


Magnets You are making your own photo magnets. Your photo is 8 inches by 10 inches. The image on the magnet is 2.8 inches by 3.5 inches. What is the scale factor of the reduction?

SolutionThe scale factor is the ratio of a side length of the magnet image to a side length of the

original photo , or 8 in.

2.8 in.. In simplest form,

the scale factor is 7 } 20 .

Example 3 Find a scale factor

1. A triangle has the vertices B(21, 21), C(0, 1), and D(1, 0). Find the coordinates of L, M, and N so that nLMN is a dilation of nBCD with a scale factor of 4. Sketch nBCD and nLMN.

L(24, 24),

x

y

1

1N

L

C

B

M

D

M(0, 4), N(4, 0)

2. In Example 3, what is the scale factor of the reduction if your photo is 4 inches by 5 inches?

7 } 10



Your Notes



You want to create a quadrilateral

x

y

1

1

P Q

R

SJ

K

L

JKLM that is similar to quadrilateral PQRS. What are the coordinates of M?

SolutionDetermine if JKLM is a dilation of PQRS by checking whether the same scale factor can be used to obtain J, K, and L from P, Q, and R.

(x, y) → (kx, ky)

P( 0, 3 ) → J( 0, 6 ) k 5 2

Q( 2, 2 ) → K( 4, 4 ) k 5 2

R( 2, 0 ) → L( 4, 0 ) k 5 2

Because k is the same in each case, the image is a dilation with a scale factor of 2 . So, you can use the scale factor to find the image M of point S.

S( 5, 5 ) → M( 2 p 5 , 2 p 5 ) 5 M( 10, 10 )

Example 4 Find missing coordinates

3. You want to create a quadrilateral QRST that is similar to quadrilateral WXYZ. What are the coordinates of T?

x

y

1

1

W

Q

R

S

Y

Z

X

T(10, 15)


Homework

Your Notes


Words to ReviewGive an example of the vocabulary word.

Ratio

4 } 3 or 4 : 3

Means

The means of a } b 5 c }

d are

b and c.

Geometric mean

The geometric mean of two positive numbers a and b is the positive number x that satisfies a } x 5 x }

b .

Scale

The scale of a map is 1 inch to 25 miles.

Scale factor of two similar polygons

A D

G

F

H

J

CB

3

6

2

4 4

12 2

The scale factor of

ABCD to FGHJ is 1 } 2 .

Golden ratio 1.618...

Proportion

x } 4 5 3 } 12

Extremes

The extremes of a } b 5 c }

d

are a and d.

Scale drawings

A scale drawing is a drawing that is the same shape as the object it represents.

A map is an example of a scale drawing.

Similar polygons

A D

G

F

H

J

CB

ABCD , FGHJ

Fibonacci sequence

1,1,2,3,5,8,13,...

174 Words to Review • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.


Golden rectangle

1

1.618…

Center of dilation

x

y

O

B C

A

X

Y Z

The center of dilation is (0, 0).

Reduction

A dilation with a scale factor greater than 0 and less than 1 is a reduction.

Dilation

x

y

O

B C

A

X

Y Z

Scale factor of a dilation

x

y

O

B C

A

X

Y Z

The scale factor of the

dilation is XY } AB

.

Enlargement

A dilation with a scale factor greater than 1 is an enlargement.

Review your notes and Chapter 6 by using the Chapter Review on pages 435–438 of your textbook.

Copyright © Holt McDougal. All rights reserved. Words to Review • Geometry Notetaking Guide 175


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