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

Centers of Triangles or Points of Concurrency


Centers of triangles or points of concurrency

Medians

Median

vertex to midpoint


Centers of triangles or points of concurrency

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 =


Centers of triangles or points of concurrency

How many medians does a triangle have?

Three – one from each vertex


Centers of triangles or points of concurrency

The medians of a triangle are concurrent.

The intersection of the medians is called the CENTRIOD.

They meet in a single point.


Centers of triangles or points of concurrency

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


In abc an bp and cm are medians

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


In abc an bp and cm are medians1

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


In abc an bp and cm are medians2

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


In abc an bp and cm are medians3

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


Centers of triangles or points of concurrency

Angle Bisector

Angle Bisector

vertex to side cutting angle in half


Centers of triangles or points of concurrency

Example 1

W

X

1

2

Z

Y


Centers of triangles or points of concurrency

Example 2

F

I

G

5(x – 1) = 4x + 1

5x – 5 = 4x + 1

x = 6

H


Centers of triangles or points of concurrency

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

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

Point P is called the __________.

Incenter


Example 4

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


Centers of triangles or points of concurrency

Perpendicular Bisector

Perpendicular Bisector

midpoint and perpendicular

(don't care about no vertex)


Centers of triangles or points of concurrency

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

NO

NO

YES


Centers of triangles or points of concurrency

Example 2: Find x

3x + 4

5x - 10

x = 7


Centers of triangles or points of concurrency

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

The Circumcenter is equidistant from the vertices of the triangle.

PA = PB = PC


Find da

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


Centers of triangles or points of concurrency

Altitude

Altitude

vertex to opposite side and perpendicular


Centers of triangles or points of concurrency

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

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

YES

NO

YES


Centers of triangles or points of concurrency

How many altitudes does a triangle have?

Three

The altitudes of a triangle are concurrent.

The intersection of the altitudes is called the ORTHOCENTER.


Centers of triangles or points of concurrency

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

NEITHER

ALTITUDE

PER. BISECTOR

BOTH


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