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Copyright © by Holt, Rinehart and Winston. 6 Holt Geometry
All rights reserved.
Name Date Class
LESSON Reteach
Perpendicular and Angle Bisectors5-1
Theorem Example
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a
segment, then it is equidistant, or the same
distance, from the endpoints of the segment.
Given: ø is the perpendicular bisector of _
FG .
Conclusion: AF 5 AG
The Converse of the Perpendicular Bisector Theorem is also true. If a point is equidistant
from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
You can write an equation for the perpendicular bisector of a segment. Consider the
segment with endpoints Q (25, 6) and R (1, 2).
Step 1 Find the midpoint of _
QR . Step 2 Find the slope of the ' bisector of _
QR .
s x1 1 x2 ______ 2 ,
y1 1 y2 ______ 2
d 5 s 25 1 1 _______ 2 , 6 1 2 _____
2 d
y2 2 y1 ______ x2 2 x1 5 2 2 6 ________
1 2 (25) Slope of
_
QR
5 (22, 4) 5 2 2 __ 3
So the slope of the ' bisector of _
QR is 3 __ 2 .
Step 3 Use the point-slope form to write an equation.
y 2 y1 5 m (x 2 x1) Point-slope form
y 2 4 5 3 __ 2
(x 1 2) Slope 5 3 __ 2 ; line passes through (22, 4), the midpoint of
_
QR .
Find each measure.
16
14
2.5
4
3 1
5 3
1. RT 5 16 2. AB 5 5 3. HJ 5 7
Write an equation in point-slope form for the perpendicular bisector
of the segment with the given endpoints.
4. A (6, 23), B (0, 5) 5. W (2, 7), X (24, 3)
y 2 1 5 3 __ 4 (x 2 3) y 2 5 5 2 3 __
2 (x 1 1)
Each point on ø is
equidistant from
points F and G.
Copyright © by Holt, Rinehart and Winston. 7 Holt Geometry
All rights reserved.
Name Date Class
LESSON Reteach
Perpendicular and Angle Bisectors continued5-1
Theorem Example
Angle Bisector Theorem
If a point is on the bisector of an angle,
then it is equidistant from the sides of
the angle.
Given:
___ › MP is the angle bisector of /LMN.
Conclusion: LP 5 NP
Converse of the Angle
Bisector Theorem
If a point in the interior of an angle is
equidistant from the sides of the angle,
then it is on the bisector of the angle.
Given: LP 5 NP
Conclusion:
___ › MP is the angle bisector of /LMN.
Find each measure.
6. EH 7. m/QRS 8. m/WXZ
23 52° 35°
Use the figure for Exercises 9–11.
9. Given that
__ › JL bisects /HJK and LK 5 11.4, find LH.
11.4
10. Given that LH 5 26, LK 5 26, and m/HJK 5 122°,
find m/LJK.
61°
11. Given that LH 5 LK, m/HJL 5 (3y 1 19)°, and
m/LJK 5 (4y 1 5)°, find the value of y.
14
Point P is equidistant
from sides
__ › ML and
___ › MN .
/LMP > /NMP