04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

Warm-UpTake out homework p.301 (4-20) evenCheck your answers with a partner

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

Median

Picture: Both sides are congruent

Median

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

Midpoint-If you have a midpoint- then the segments on both sides are CONGRUENT! That is why you will see the “tick marks”For the measure of the entire line- add both sides!

M D P

N

C

In MC and ND are medians.MNP,

1. What is NC if NP = 18?

9

2. If DP = 7.5, find MP. 15

A B C

E

D

1.What is ED if DC = 14?

2.What Is AC is BC is 9?

14

3.If BC = 3, find AC.

6

AD and EB are medians:

So if you are given the length of the entire side, how do you find a missing segment?If you are given the length of a segment, how to you find the entire side?If you have a median- what do you know about each side?

A E B

CD

If CD = 2x + 5, BD = 4x – 1, and AE = 5x –2, find BE.

In CE and AD are medians.ABC,

BD = CD 4x – 1= 2x + 5

2x = 6 x = 3

AE = BEBE = 5x – 2BE = 5(3) – 2

BE = 13

The intersection of the medians is called the CENTROID.

How many medians does a triangle have?

Draw the Picture:

When three or more lines or segments meet at the same point.

Concurrent:

Theorem 6-1

Distance from vertex to

centroid is twice the distance from

centroid to midpoint.

Draw Picture

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

Vertex to Centroid LONGER Centroid to Midpoint shorter

x + 2x = whole median

A BF

X

E

C

D

Quick AssessmentWhat is a median?What is a centroid?What does concurrent mean?What is the vertex?What is a midpoint?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

A BF

X

E

C

D

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

In ABC, AN, BP, and CM are medians.

A

B

M

P E

C

N

If EM = 3, find EC.

EC = 2(3)

Ex: 1

EC = 6

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

In ABC, AN, BP, and CM are medians.

A

B

M

P E

C

NIf EN = 12, find AN.AE = 2(12)=24

Ex: 2

AN = 36

AN = AE + ENAN = 24 + 12

Altitude

Picture:

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

Altitude

The altitude is the “true

height” of the triangle.

Tell whether each red segment is an altitude of the triangle.The altitude is the “true height” of

the triangle.

Perpendicular Bisector

Picture:

Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector

Tell whether each red segment is an perpendicular bisector of the triangle.

Can you have both?Can it be both an altitude and perpendicular bisector?

Quick AssessmentWhat is the difference between altitude and perpendicular bisector?What is an altitude?What is a perpendicular bisector?

How can the segment be both- altitude and perpendicular bisector?

04/19/23

EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

Angle bisector of a triangle : is the bisector of each angle of the triangle. (cuts each angle in

half)

Incenter of the triangle: where the angle bisectors meet.

Angle Bisectors

Incenter

Concurrency of AngleBisectors of a Triangle

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

B

F

A

D

P

E

C

PD = PE = PF