Centers of triangles or points of concurrency
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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.

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Centers of Triangles or Points of Concurrency

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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


The incenter is the same distance from the sides of the triangle.

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

(don't care about no vertex)


Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.

NO

NO

YES


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.


The Circumcenter is equidistant from the vertices of the triangle.

PA = PB = PC


Example 3: The perpendicular bisectors of triangle ABC meet at point P.

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


Tell whether each red segment is an altitude of the triangle.

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


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