5.1

Perpendicular and Angle Bisectors

Definitions• Distance from a point to a line:

• Length of the perpendicular segment from a point to a line.

• Equidistant: • A point that is the same distance from two

segments (rays or lines).

Angle Bisector Theorem• If a point is on the bisector of an angle, then it is

equidistant from the two sides of the angle.

A

B

C

D

If bisects ABC and

D is a point on ,

then

BD

BD

DA DC

Example #1

) If bisects HJK and LK = 11.4, find LH.

b) If LH = 26, LK = 26 and m HJK= 122 , find m LJK.

c) If LH = LK, m HJL = (3y +19) and

m LJK = (4y +5) , find the value of y.

a JL

L

HJ

K

Defn: Perpendicular Bisector• A line that is perpendicular to and bisects another segment.

A

B

C D

is the perpendicular bisector of AB CD�

Perpendicular Bisector Theorem• If a point is on the perpendicular bisector of a segment,

then it is equidistant from the endpoints of the segment.

If is the bis of , ...AB CD then�

E

B

C D

A

Example #2 & 3 is the bisector of each triangle.

Find the value of x or y.MY �

A

M

BY

3 4x 2 10x

6x

M Y

D

C

3 2x y

5x y

2 9x

4 3x

Example #4 Given: is the bisector of Find: x and y

AH MTA

M TH

2x + 3 y + 4

½ x + 8 3y

Write eqn of perp bisector given two endpoints• STEPS:

• 1) Find the midpoint of the segment• 2) Find the slope of the segment• 3) Find the slope of its perpendicular line• 4) Use the perpendicular slope and midpoint and plug

into Point-Slope form

5) A ( 6, -3) B (0,5) 6) A (2, 7) B (-4, 3)