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Centers of Triangles or Points of ConcurrencyPowerPoint Presentation

Centers of Triangles or Points of Concurrency

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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.

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

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

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

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

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 triangle.

8

D

F

L

C

B

E

Example 4The 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 triangle.

Perpendicular Bisector

midpoint and perpendicular

(don't care about no vertex)

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

NO

NO

YES

How many perpendicular bisectors does a triangle have? triangle.

Three

The perpendicular bisectors of a triangle are concurrent.

The intersection of the perpendicular bisectors is called the CIRCUMCENTER.

The triangle.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

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

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

YES

NO

YES

How many altitudes does a triangle have? at point P.

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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