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2.1b Triangle Properties

-Special Segments in Triangles

CCSS

G-CO.10 Prove theorems about triangles. Theorems

include measures of interior angles of a

triangle sum to 180 base angles of isosceles

triangles are congruent the segment joining

midpoints of two sides of a triangle is parallel

to the third side and half the length the

medians of a triangle meet at a point.

GSEs

M(GM)102 Makes and defends conjectures,

constructs geometric arguments, uses geometric

properties, or uses theorems to solve problems

involving angles, lines, polygons, circles, or

right triangle ratios (sine, cosine, tangent)

within mathematics or across disciplines or

contexts

Median

- Connects a vertex to the of the

opposite side

MIDDLE

B

Tells us that it cut the side BC in half so BD

DC

F

D

When you combine ALL the medians of one triangle,

you get the CENTROID

A

C

E

A centroid would balance the triangle if you held

it up with a pencil

Example

Determine the coordinates of J so that SJ is a

median of the triangle.

Ans Use the midpoint formula for GB

J

J ( 9, 0.5)

J

J

Altitude

- Connects the vertex to where it is perpendicular

to the opposite side

Combine all three ALTIUDES in a triangle and you

get a ORTHOCENTER

A

Z

T

W

R

P

Tells us the segment is perpendicular

Altitude with an obtuse triangle

Example

BD is an altitude of Triangle ABC

Find BC, and AC

Ans 7x2090

7x 70

3x-5

X 10

But wait.. Re-read the question

Find BC, so BC 3(10)-5

25

Angle Bisector

- A segment that cuts one vertex angle in half and

goes to the opposite side of the triangle

Draw angle bisector AF

B

Indicates the angle A was bisected

F

A

C

Example

N

Find m NWB if WT is an angle bisector Of

WNB

T

m NWT 3x 8 m NWB 3x 34

3x8

ANS Draw your diagram

W

3x8 3x8 3x 34 6x 16 3x 34 3x

18 x 6 3(6) 34 52

B

Perpendicular Bisector

- A segment that
- 1) is perpendicular to one side of

the triangle - 2) bisects the same side

H

Draw the segment AT that is a perpendicular

bisector of BO on Triangle HBO

B

O

example

- Name all segments that are (if any)
- Angle Bisectors
- Perpendicular Bisectors
- Altitudes
- Medians

QU

NONE

RT

SP

Example

- has vertices A(-4,1) , B ( 1, 6) and

C (3, -4). - Find 1) The coordinate of T if it is on AB and

TR is a perpendicular bisector to side AB - 2) Find the slope of TR

Activity

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- End of notes