Unit 5: Relationships of Triangles

Perpendiculars and Bisectors

Objectives: SWBAT use properties of perpendicular bisectors. SWBAT use properties of angle bisectors to identify equal distances.

Perpendicular Bisector

A segment that is perpendicular and bisects a segment

Equidistant The same distance

Perpendicular Bisector Theorem A perpendicular Bisector is equidistant from the other angles of

A triangle.

Converse of Perpendicular Bisector Theorem If a segment is equidistant from the other two angles of a triangle, and goes through the midpoint of that side, then it is a perpendicular bisector.

Tell whether the information in the diagram allows you to conclude that C is on the

perpendicular bisector of AB . If so, find the missing variable.

Must be two of the three; equidistant from the corners, a bisector, or form a right

angle to be a perpendicular bisector.

1. 2. 3.

4 1

Yes

x .Not

Enough

Info

56 34 90

4 7 2 3

2 7 3

2 10

5

Yes

( ) It ist at Right

x x

x

x

x

Find the measure of the following. 4. x 5. AD 6. BA

Angle Bisectors A segment that bisects (cuts in half) an angle

Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Converse of the Angle Bisector Theorem If a point is in the interior of an angle is equidistant from the sides of an angle, then

it is on the bisector of that angle.

5 12 8

12 3

4

x x

x

x

2 5 10

5 10

5

5 10 15

x x

x

x

AD

2

2

5 14

5 14 0

7 2 0

7 0 2 0

7 2

14

x x

x x

x x

x x

x x

Both Work

BA

Tell whether the information in the diagram allows you to conclude that P is on the bisector of A. Explain.

The ray in the interior needs to be equidistant (meaning there are tick marks / data as well as a perpendicular sign), then it is an angle bisector.

7. 8. 9.

Find the measure of the following.

10. x 11. 𝒎∠𝑿𝒁𝑾 12. 𝒎∠𝑴𝑳𝑷

Finding Perpendicular Bisectors on a Coordinate Plane

1. Find the midpoint of the two points

2. Find the slope Perpendicular to the given points

3. Put the data into 𝒚 = 𝒂(𝒙 − 𝒉) + 𝒌, and simplify

KP MP

YES

Not

Enough

Info

( no tick marks )

YES

3 2 1

3 1

2

x x

x

x

34

It' s an Bi sector

m XZW m WZY

22 22

44

It' s an Bi sec tor

m MLP m MNL m PNL

x

x

13. A segment has endpoints of 𝑨(𝟐, 𝟒) and 𝑩(𝟔, 𝟎). Find the equation of the perpendicular bisector of 𝑨𝑩̅̅ ̅̅

14. A segment has endpoints of 𝑻(−𝟒, 𝟓) and 𝑺(𝟔, 𝟏). Find

the equation of the perpendicular bisector of 𝑺𝑻̅̅̅̅

2 1

2 1

0 4 41

6 2 4

1

Slope

y ym

x x

m

Opposite Recipricol

m

1 2 1 2

2 2

2 6 4 0

2 2

8 4

2 2

4 2

Midpoint

x x y ymidpt , ,

midpt ,

midpt ,

midpt ,

h,k

1 4 2

1 4 2

2

Equation

y a x h k

y x

y x

y x

2 1

2 1

1 5 4 2

6 4 10 5

5

2

Slope

y ym

x x

m

Opposite Recipricol

m

1 2 1 2

2 2

4 6 5 1

2 2

2 6

2 2

1 3

Midpoint

x x y ymidpt , ,

midpt ,

midpt ,

midpt ,

h,k

5

1 32

5 53

2 2

5 1

2 2

Equation

y a x h k

y x

y x

y x

Points of Concurrency

Objectives: SWBAT use properties of perpendicular bisectors of a triangle. Objectives: SWBAT use properties of angle bisectors of a triangle.

Perpendicular Bisector of a Triangle A segment that is perpendicular and bisects a side of a Triangle.

Concurrent Lines

Lines that intersect

Point of Concurrency

The point where three lines intersect Concurrency with triangle bisectors (Location of Points of Concurrency)

Acute Triangle Right Triangle Obtuse Triangle

Inside the Triangle On the Triangle Outside the Triangle

Circumcenter of the Triangle The circumcenter is the point where the 3 perpendicular

Bisectors of a triangle intersect.

Perpendicular Bisectors of a Triangle Theorem

The circumcenter is equidistant from the Vertices of a triangle.

1. The perpendicular bisectors of ABC meet at point G.

Because we have Right Triangles, we will see a lot of Pythagorean Theorem.

a. Find 𝐺𝐶

b. Find 𝐵𝐺

c. Find 𝐹𝐶

7GE GA GB

7GE GA GB

2 2 2

2 2 2

2

2

2 7

4 7

3

3

a b c

x

x

x

x

2. Use the diagram shown. D is the circumcenter of ABC. 𝐷𝐶 = 3 and 𝐷𝐹 = √3

a. Find the length of DA.

b. Find the length of AB

c. Find the length of FA.

d. Explain why ADF BDE.

Angle Bisector of a triangle

Segments that bisect the angles of triangle.

Incenter

The point of concurrency of all the angle bisectors of a triangle.

3DA DC DB

2 2 2

2 2 2

2

2

3 3

3 9

6

6

a b c

x

x

x

x

3 3

6

AB BD DA

AB

AB

HL

Angle Bisectors of a Triangle Theorem The Incenter is equidistant from the sides of a triangle.

3. The angle bisectors of MNP meet at point L. 𝐿𝑁 = 17, 𝑎𝑛𝑑 𝑄𝑁 = 15

a. Label all congruent angles.

b. Find segments that are congruent.

c. Find LQ and RN.

SML QML

SPL RPL

RNL QNL

SL LR LQ

2 2 2

2 2 2

2

2

15 17

225 289

64

8

a b c

x

x

x

x

2 2 2

2 2 2

2

2

8 17

64 289

225

15

a b c

y

y

y

y

4. Use the diagram shown. V is the Incenter of XWZ , VT = 3, m WXT 20 ,

XW WZ

a. Find the length of VS.

b. Find the m SZX .

c. Explain why XSV ZTV .

3VS VY VT

20

It is an Iso.

Base s are

AAS

Medians of a Triangle

Objectives: SWBAT use properties of medians of a triangle.

Median

A segment from a vertex that bisects the opposite side.

Centroid The intersection of the 3 medians of a triangle.

Medians of Triangles Theorem

The centroid of a triangle is two-thirds of the distance

from each vertex to the midpoint of the opposite side.

P is the centroid of QRS shown.

Use your eyes to see which part of the median being used is longer (that will be the 2/3) piece.

1. If RT = 15, find RP.

2. If QU = 27, find PU.

3. If PS = 8, find VS and VP.

2

3

215

3

10

RP RT

RP

RP

1

3

127

3

9

PU QU

PU

PU

1

3

18

3

24

PS VS

VS

VS

2

3

224

3

16

VP VS

VP

VP

Find the Following

NS

QT

QR

MU

L is the centroid of MNO , NP = 11, ML = 10 and NL = 8

4. Find the length of PO .

5. Find the length of MP .

6. Find the length of LQ .

7. Find the length of NQ .

8. Find the perimeter of NLP

Given: S is the centroid of MQU

27

10

8

9

NU

ST

RU

MT

11

P is the midpoint

NP PO

PO

210

3

210

3

15

MP

MP

MP

1

2

18

2

4

LQ LN

LQ

LQ

8 4

12

NQ NL LQ

NQ

NQ

8 5 11

27

Peri NL LP NP

18

8

9

30

Q

N

QT

R

S

MU

Finding the Centroid using Coordinate system

1. Graph the 3 points of a triangle

2. Find the midpoint of any side, and sketch the median

3. Find 2/3 of the distance between the vertex and the midpoint

The vertices of ∆𝑭𝑮𝑯 are 𝑭(𝟐, 𝟑), 𝑮(𝟒, 𝟖) and 𝑯(𝟔, 𝟏). Find the coordinate of the centroid.

The vertices of ∆𝑭𝑮𝑯 are 𝑭(−𝟑, 𝟑), 𝑮(𝟏, 𝟓) and 𝑯(−𝟏, −𝟐). Find the coordinate of the

centroid.

Altitudes of a Triangle

Objectives: SWBAT use properties of altitudes of a triangle.

Altitude

Orthocenter

Altitudes of Triangles Theorem

Examples

Use the diagram shown and the given information to decide in each case whether AD is a

perpendicular bisector, an angle bisector, a median or an altitude of ABC.

1. DB DC

2. BAD CAD

3. DB DC and AD BC

4. AD BC

5. BAD CAD

A

B

C

D

Finding Altitudes on a Coordinate Plane

6. A triangle has endpoints of 𝑨(𝟐, 𝟒), 𝑩(𝟔, 𝟎) and 𝑪(𝟎, 𝟎). If 𝑪𝑴̅̅ ̅̅ ̅ is the altitude of ∆𝑨𝑩𝑪, what is the coordinates of 𝑴?

7. A triangle has endpoints of 𝑱(−𝟓, −𝟑), 𝑲(𝟑, 𝟗) and 𝑳(𝟕, 𝟐). If 𝑳𝑴̅̅̅̅̅ is the altitude of ∆𝑱𝑲𝑳, what is the coordinates of 𝑴?

Statements Reasons

Given: CFD EFD

FD is an altitude

Prove: FD is a median

Inequalities in Triangles

Objectives: SWBAT use triangle measurements to decide which side is longest or which

angle is largest. SWBAT use the triangle inequality.

Exterior Angle Theorem

Review: Solve for the variable.

1. 2.

Exterior Angle Inequality Theorem

Complete the following inequalities based on the information below. 3. _____m S m TRU 4. _____m PNO m PON 5. _____m BAC m BCA

Comparing Triangles

Largest Angle

Shortest

side

Longest Side Smallest

angle

Side vs. Angle Theorem

1.

2.

Name the shortest and longest sides of the triangle.

1. 2.

Name the smallest and largest angles of the triangle.

3. 4. 5.

List all the angles and sides in ascending order.

6. 7.

Triangle Inequality Theorem

Is it possible to have the following dimensions for triangle ABC

1. AB = 8, BC = 15, and AC = 17 2. AB = 6, BC = 8, and AC = 14

3. AB = 1, BC = 1, and AC = 2 4. AB = 5, BC = 5, and AC = 5

Find the possible measures for XY in XYZ .

1. XZ = 2 and YZ = 3 2. XZ = 8 and YZ = 10 3. XZ = 7 and YZ = 11

Hinge Theorem

Hinge Theorem Converse

Complete with <, >, or =, if possible.

1. 𝐴𝐵 _______ 𝐹𝐸 2. 𝑃𝑇 _______ 𝑋𝑌 3. 𝐽𝐾 _______ 𝐾𝐿

4. m 1_______ m 2 5. m ABD _______ m CBD 6. m 1____ m 2

The angles a trinagle cannot ever be _______________________________. Review: Solve the following compound inequalities.

8. 𝟏𝟓 ≤ 𝟐𝒙 + 𝟑 ≤ 𝟑𝟔 9. 𝟎 ≤𝟓𝒙−𝟏𝟓

𝟐≤ 𝟗𝟎

Write the inequality for the range for the variable. 10. 11.