Centers of Triangles or Points of Concurrency

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# Centers of Triangles or Points of Concurrency - PowerPoint PPT Presentation

Centers of Triangles or Points of Concurrency. Medians. Median. vertex to midpoint. Example 1. M. D. P. C. What is NC if NP = 18?. MC bisects NP…so 18/2. 9. N. If DP = 7.5, find MP. 15. 7.5 + 7.5 =. How many medians does a triangle have?. Three – one from each vertex.

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### Centers of Triangles or Points of Concurrency

Medians

Median

vertex to midpoint

Example 1

M

D

P

C

What is NC if NP = 18?

MC bisects NP…so 18/2

9

N

If DP = 7.5, find MP.

15

7.5 + 7.5 =

How many medians does a triangle have?

Three – one from each vertex

The medians of a triangle are concurrent.

The intersection of the medians is called the CENTRIOD.

They meet in a single point.

Theorem

The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

2x

x

Example 2

In ABC, AN, BP, and CM are medians.

If EM = 3, find EC.

C

EC = 2(3)

N

P

E

EC = 6

B

M

A

Example 3

In ABC, AN, BP, and CM are medians.

If EN = 12, find AN.

C

AE = 2(12)=24

AN = AE + EN

N

P

E

AN = 24 + 12

B

AN = 36

M

A

C

N

P

E

B

M

A

Example 4

In ABC, AN, BP, and CM are medians.

If EM = 3x + 4 and CE = 8x, what is x?

x = 4

C

N

P

E

B

M

A

Example 5

In ABC, AN, BP, and CM are medians.

If CM = 24 what is CE?

CE = 2/3CM

CE = 2/3(24)

CE = 16

Angle Bisector

Angle Bisector

vertex to side cutting angle in half

Example 1

W

X

1

2

Z

Y

Example 2

F

I

G

5(x – 1) = 4x + 1

5x – 5 = 4x + 1

x = 6

H

How many angle bisectors does a triangle have?

three

The angle bisectors of a triangle are ____________.

concurrent

The intersection of the angle bisectors is called the ________.

Incenter

Point P is called the __________.

Incenter

A

8

D

F

L

C

B

E

Example 4

The angle bisectors of triangle ABC meet at point L.

• What segments are congruent?
• Find AL and FL.

LF, DL, EL

Triangle ADL is a right triangle, so use Pythagorean thm

AL2 = 82 + 62

AL2 = 100

AL = 10

FL = 6

6

Perpendicular Bisector

Perpendicular Bisector

midpoint and perpendicular

Example 2: Find x

3x + 4

5x - 10

x = 7

How many perpendicular bisectors does a triangle have?

Three

The perpendicular bisectors of a triangle are concurrent.

The intersection of the perpendicular bisectors is called the CIRCUMCENTER.

PA = PB = PC

Find DA.

DA = 6

BA = 12

• Find BA.
• Find PC.

PC = 10

• Use the Pythagorean Theorem to find DP.

B

6

DP2 + 62 = 102

DP2 + 36 = 100

DP2 = 64

DP = 8

10

D

P

A

C

Altitude

Altitude

vertex to opposite side and perpendicular

The altitude is the “true height” of the triangle.

YES

NO

YES

How many altitudes does a triangle have?

Three

The altitudes of a triangle are concurrent.

The intersection of the altitudes is called the ORTHOCENTER.

Tell if the red segment is an altitude, perpendicular bisector, both, or neither?

NEITHER

ALTITUDE

PER. BISECTOR

BOTH