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This ppt includes the following topics:-

Medians

Centroid

Angle Bisector

Incentre

Altitude

Orthocentre

Perpendicular Bisector and many more.

Medians

Centroid

Angle Bisector

Incentre

Altitude

Orthocentre

Perpendicular Bisector and many more.

1.
Centers of Triangles or

Points of Concurrency

Prepared for Ms. Pullo’s

Geometry Classes

Points of Concurrency

Prepared for Ms. Pullo’s

Geometry Classes

2.

3.
Example 1

In MNP, MC and ND are medians.

M D P

C What is NC if NP = 18?

MC bisects NP…so 18/2 9

If DP = 7.5, find MP.

7.5 + 7.5 = 15

In MNP, MC and ND are medians.

M D P

C What is NC if NP = 18?

MC bisects NP…so 18/2 9

If DP = 7.5, find MP.

7.5 + 7.5 = 15

4.
How many

medians does a

triangle have?

medians does a

triangle have?

5.
The medians of

a triangle are

concurrent.

The intersection of

the medians is

called the

CENTRIOD.

a triangle are

concurrent.

The intersection of

the medians is

called the

CENTRIOD.

6.
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.

The length of the segment

from the vertex to the

centroid is twice the length

of the segment from the

centroid to the midpoint.

7.
Example 2

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

If EM = 3, find C

EC = N

P

2(3) E

EC = 6 B

M

A

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

If EM = 3, find C

EC = N

P

2(3) E

EC = 6 B

M

A

8.
Example 3

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

If EN = 12, find

AE = C

AN = AE + EN

2(12)=24 N

P E

AN = 24 +

B

12

AN = A M

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

If EN = 12, find

AE = C

AN = AE + EN

2(12)=24 N

P E

AN = 24 +

B

12

AN = A M

9.
Example 4

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

If EM = 3x + 4

and CE = 8x, C

what is x?

N

P E

x=4 M

B

A

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

If EM = 3x + 4

and CE = 8x, C

what is x?

N

P E

x=4 M

B

A

10.
Example 5

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

If CM = 24

what is CE? C

CE = 2/3CM N

CE = P E

2/3(24) B

CE = 16 M

A

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

If CM = 24

what is CE? C

CE = 2/3CM N

CE = P E

2/3(24) B

CE = 16 M

A

11.
Angle Bisector

12.
Example 1

In WYZ, ZX bisects WZY . If m1 = 55,

find mWZY .

W mWZY 55 55

mWZY 110

X

1

2

Z Y

In WYZ, ZX bisects WZY . If m1 = 55,

find mWZY .

W mWZY 55 55

mWZY 110

X

1

2

Z Y

13.
Example 2

In FHI, IG is an angle bisector. Find mHIG.

F

5( x 1) I

G (4 x 1)

5(x – 1) = 4x + 1

H 5x – 5 = 4x + 1

x=6

In FHI, IG is an angle bisector. Find mHIG.

F

5( x 1) I

G (4 x 1)

5(x – 1) = 4x + 1

H 5x – 5 = 4x + 1

x=6

14.
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

does a triangle have?

three

The angle

bisectors of a

triangle are

concurrent

____________.

The intersection of the

angle bisectors is

called the ________.

Incenter

15.
The incenter is the same distance from the

sides of the triangle.

Point P is called

B

the __________.

Incenter

D

F P

A E C

sides of the triangle.

Point P is called

B

the __________.

Incenter

D

F P

A E C

16.
Example 4

The angle bisectors of triangle ABC meet at point L.

• What segments are congruent? LF, DL, EL

• Find AL and FL.

Triangle ADL is a

A right triangle, so use

FL = 6 Pythagorean thm

8

D AL2 = 82 + 62

F AL2 = 100

L AL = 10

6

C E B

The angle bisectors of triangle ABC meet at point L.

• What segments are congruent? LF, DL, EL

• Find AL and FL.

Triangle ADL is a

A right triangle, so use

FL = 6 Pythagorean thm

8

D AL2 = 82 + 62

F AL2 = 100

L AL = 10

6

C E B

17.

18.
Tell whether each red segment is an altitude of the

The altitude is the

“true height” of

the triangle.

The altitude is the

“true height” of

the triangle.

19.
How many altitudes

does a triangle have?

The altitudes of

a triangle are

concurrent.

The intersection of the

altitudes is called the

ORTHOCENTER.

does a triangle have?

The altitudes of

a triangle are

concurrent.

The intersection of the

altitudes is called the

ORTHOCENTER.

20.
Perpendicular Bisector

21.
Example 1: Tell whether each red segment is

a perpendicular bisector of the triangle.

a perpendicular bisector of the triangle.

22.
Example 2: Find x

3x + 4 5x - 10

3x + 4 5x - 10

23.
How many perpendicular

bisectors does a triangle

have?

The perpendicular

bisectors of a triangle

are concurrent.

The intersection of the

perpendicular bisectors is called

the CIRCUMCENTER.

bisectors does a triangle

have?

The perpendicular

bisectors of a triangle

are concurrent.

The intersection of the

perpendicular bisectors is called

the CIRCUMCENTER.

24.
The Circumcenter is

equidistant from the vertices

of the triangle.

B

PA = PB = PC

P

A C

equidistant from the vertices

of the triangle.

B

PA = PB = PC

P

A C

25.
Example 3: The perpendicular bisectors of

triangle ABC meet at point P.

• Find DA. DA = 6

• Find BA. BA = 12

• Find PC. PC = 10

• Use the Pythagorean Theorem B

to find DP.

DP2 + 62 = 102 6

D 10

DP + 36 = 100

2

DP2 = 64 P

A C

DP = 8

triangle ABC meet at point P.

• Find DA. DA = 6

• Find BA. BA = 12

• Find PC. PC = 10

• Use the Pythagorean Theorem B

to find DP.

DP2 + 62 = 102 6

D 10

DP + 36 = 100

2

DP2 = 64 P

A C

DP = 8

26.
Tell if the red segment is an altitude,

perpendicular bisector, both, or neither?

NEITHER

ALTITUDE

PER.

BOTH BISECTOR

perpendicular bisector, both, or neither?

NEITHER

ALTITUDE

PER.

BOTH BISECTOR

27.
IN A NUT SHELL

Median – Centroid

Angle Bisector – Incenter

Altitude – Orthocenter

Perpendicular Bisector - Circumcenter

Angle Bisector: The Incentor is equidistance to

the sides

Perpendicular Bisector – the Circumcenter is

equidistance to the

vertex

Median – Centroid

Angle Bisector – Incenter

Altitude – Orthocenter

Perpendicular Bisector - Circumcenter

Angle Bisector: The Incentor is equidistance to

the sides

Perpendicular Bisector – the Circumcenter is

equidistance to the

vertex

28.
The End