5.1 Perpendiculars and Bisectors

Day 1 Part 1

CA Standard 16.0

Warmup

Simplify. 1. 6x + 11y – 4x + y 2. -5m + 3q + 4m – q 3. -3q – 4t – 5t – 2p 4. 9x – 22y + 18x – 3y 5. 5x2 + 2xy – 7x2 + xy

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB,

then CA = CB.

A P B

C

Converse of the Perpendicular Bisector Theorem

In the diagram shown, MN is the perpendicular bisector of ST.

What segment lengths in the diagram are equal?

Explain why Q is on MN

T

M N

S

Q

12

12

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If m<BAD = m<CAD, then DB = DC.

A

B

C

D

Converse of the Angle Bisector Theorem

Use the diagram to answer the following. In the diagram, F is on the bisector of

< DAE. If m<BAF = 50, then m<CAF = ____ If FC = 10, then FB = ____ Is triangle ABF congruent to triangle ACF?

Explain. A

B

D

G

F

C

E

5.2 Bisectors of a Triangle

Day 1 Part 2

CA Standards 16.0, 21.0

In the figure, YW bisects <XYZ.

m<XYZ = 6x + 2, m<ZYW = 8x – 6.

Solve for x and find m<XYZ.

Y

X

W

Z

Concurrency of Perpendicular Bisectors of a

Triangle The perpendicular bisectors of a triangle

intersect at a point

that is equidistant from

the vertices of the

triangle.

PA = PB = PCA

B

C

P

P is also called the circumcenter of the triangle.

Use the diagram shown.

E is the circumcenter of Δ ABC. DA = ___ BF = ___ <EFC = ___

A

D

B

E

CF

Definitions

Concurrent lines: three or more lines intersect in the same point.

Point of concurrency: the point of intersection of the lines.

Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

PD = PE = PF

A

B

C

P

E

D F

The point of concurrency can be inside the triangle, on the triangle, or outside of the triangle. Acute Triangle: inside Right Triangle: on Obtuse Triangle: outside

Example

Which segments

are congruent?

M

P N

RQ

L

S

Use the diagram shown.

E is the circumcenter of Δ ABC. DA = ___ BF = ___ <EFC = ___

A

D

B

E

CF

Mini quiz on definitions…

The _____________ of the angle bisectors is called the incenter of the triangle.

If three or more lines intersect at the same point, the lines are ________.

The point of concurrency of the perpendicular bisectors of a triangle is called ____________________.

Point of concurrency

Concurrent

Circumcenter of the triangle

Construction

Pg. 268 # 14, 15 Pg. 275 # 5 – 9

Pg. 269 # 21 – 29, 32 Pg. 275 # 10 - 22

5.3 Medians and Altitudes of a Triangle

Day 2 Part 1

CA Standards 16.0

Warmup

Find BD.

12 12

15

C

D

B A

AC = ___ m<DCB = ___ m<B = ___

A

B

C20

L

D

55

20

55

35

Median of a triangle.

Median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

Median

The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle.

•P

Centroid

Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

AP = 2/3 AD

BP = 2/3 BF

CP = 2/3 CE

A B

C

DF

E

P

P is the centroid of ∆QRS shown. Find RT and RP when PT = 5.

R

SQ T

P

Sketch ∆JKL with J(7,10), K(5,2), L(3,6). Find the coordinates of the centroid of ∆JKL.

Altitude of a triangle

An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.

Altitude

Every triangle has three altitudes.

The lines containing the altitudes are concurrent and intersect at a point: orthocenter of the triangle.

Where is the orthocenter located in each type of triangle? Acute triangle Right triangle Obtuse triangle

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

perpendicular bisector, an angle bisector, a median, or an altitude of Δ DEF.

1. DG = FG

2. EG DF

3. m<DEG = m<FEGT

E

D G F

Median

Altitude

Angle bisector

The angle bisectors of Δ ABC meet at point D. Find DE.

A

B

C

D

E F

G

LL19

28

19

5.4 Midsegment Theorem

Day 2 Part 2

CA Standards 17.0

Review

Given PQ = 14, SU = 6, and QU = 3, find the perimeter of Δ STU.

Q

U

R

S

T P

Midsegment Theorem

The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long.

DE ll AB and DE = ½ AB

A B

C

D E>

>

UW and VW are midsegment of Δ RST. Find UW and RT.

R

S

V

U

W

T812 6

16

GH, HJ and JG are midsegments of Δ EDF.

1. JH ll ___

2. EF = ___

3. DF = ___

4. ___ ll DE

5. GH = ___

6. JH = ___

D

G

F

H

EJ

24

810.6

DF

21.2

16

GH

12

8

Given the midpoints of a triangle are (7,4), (5,6) and (8,7), find the coordinates of the vertices.

Pg. 282 # 3 – 12 Pg. 283 # 17 – 20 Pg. 290 # 3 – 22, 26 – 29

5.5 Inequalities in One Triangle

Day 3 Part 1

CA Standards 6.0, 13.0

Warmup

Solve the inequality. 1. -x + 2 > 7 2. c – 18 < 10 3. -5 + m < 21 x – 5 > 4 z + 6 > -2

Theorems

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

m<B > m<CA

B

C

3 7

List the angles in order from greatest to least.

A

B C

27

23

18

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

D

E

F

60°

40°

EF > DF

Write the measurements of the triangles in order from least to greatest.

J

H

G

100°

35°

45°

Q

R

P

5

6

7

JG, HJ, HG m<R, m<Q, m<P

List the sides in order from longest to shortest.

G E

F

45° 65°

Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent (not next to) interior angles.

A

B C1

m< 1 > m<A

m<1 > m<B

Constructing a Triangle

Construct a triangle with the given group of side length, if possible. 4 in, 4 in, 4 in

2 in, 4 in, 6 in

3 in, 4 in, 5 in

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AB + BC > AC

AC + BC > AB

AB + AC > BCA

B

C

Use the diagram to solve the inequality

AB + BC > AC.

A

B

C

6x + 3

4x + 5

3x + 2

Two sides of a triangle have lengths 4 and 14. Describe the possible length of the third side.

Two sides of a triangle have lengths 10 and 15. Describe the possible length of the third side.

5.6 Indirect Proof and Inequalities in Two Triangles

Day 3 Part 2

CA Standards 2.0, 6.0, 13.0

Review

1. What is the sum of x and y?

2. Which measure is greater, x ° or y °?

x

y21

20

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

R

S

T

W

V

X100° 80°

RT > WX

Use Hinge theorem to complete the blank with <, >, or =.

1. RS __ TU 2. m<1 ___ m< 2

3. XY __ ZY

R

S

T

U110°

130°

1

2

15 13

< >

X

Y

Z

41°

38°

<

Converse of the Hinge Theorem

What is the largest angle that is part of a triangle?

9910

44

W

XY Z

Review List the sides in order from shortest to

longest.L

M

N

75°

74°

A

B

C50°

49°

75° + 74°+ m<N = 180°149° + m<N = 180°m<N = 31°

31°

LM, LN, MN

49 ° + 50 ° + m<B = 180 °99 ° + m<B = 180 °m<B = 81 °

81 °

BC, AB, AC

Review

Solve for a

2a + 7

a + 19

Extra Credit!!

Use the diagram to solve.

1. Find the value of x.

2. Find m<B

3. Find m<C

4. Find m<BAC

D

A

BC

3x°

(x+13)°(x+19)°

Pg. 298 # 6 – 25, 34 Pg. 305 # 3 – 23, 26, 27

Ch 5 Review

Day 4 Part 1

Warmup

Review

Find the value of x.

40

x

2x+6

Review

Ch 5 Culminating Task

Ch 5 Test

Day 4 Part 2