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Copyright © by Holt, Rinehart and Winston. 6 Holt Geometry All rights reserved. Name Date Class LESSON Reteach Perpendicular and Angle Bisectors 5-1 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: is the perpendicular bisector of _ FG . Conclusion: AF = 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 (-5, 6) and R (1, 2). Step 1 Find the midpoint of _ QR . Step 2 Find the slope of the bisector of _ QR . x 1 + x 2 ______ 2 , y 1 + y 2 ______ 2 = -5 + 1 _______ 2 , 6 + 2 _____ 2 y 2 - y 1 ______ x 2 - x 1 = 2 - 6 ________ 1 - (-5) Slope of _ QR = (-2, 4) = - 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 - y 1 = m (x - x 1 ) Point-slope form y - 4 = 3 __ 2 (x + 2) Slope = 3 __ 2 ; line passes through (-2, 4), the midpoint of _ QR . Find each measure. 16 14 2.5 4 3 1 5 3 1. RT = 16 2. AB = 5 3. HJ = 7 Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 4. A (6, -3), B (0, 5) 5. W (2, 7), X (-4, 3) y - 1 = 3 __ 4 (x - 3) y - 5 = - 3 __ 2 (x + 1) Each point on is equidistant from points F and G.

5-1 Perpendicular and Angle Bisectors - Dragonometrydragonometry.net/geometry/xch05/ch05-1_reteach.pdfLESSON Reteach 5-1 Perpendicular and Angle Bisectors Theorem Example Perpendicular

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Page 1: 5-1 Perpendicular and Angle Bisectors - Dragonometrydragonometry.net/geometry/xch05/ch05-1_reteach.pdfLESSON Reteach 5-1 Perpendicular and Angle Bisectors Theorem Example Perpendicular

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.

Page 2: 5-1 Perpendicular and Angle Bisectors - Dragonometrydragonometry.net/geometry/xch05/ch05-1_reteach.pdfLESSON Reteach 5-1 Perpendicular and Angle Bisectors Theorem Example Perpendicular

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