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Objective – To prove and apply theorems involving angle bisectors and perpendicular bisectors.
Perpendicular Bisector Theorem
Locus - A set of points that satisfies a given condition
If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpointsof the segment.
The locus of points equidistant from the endpoints of a segment.
A B
Perpendicular Bisector -
Given: Line is bisector of ACB is any point on Line Prove: AB BC
nn
Statement1) Draw auxillary lines AB & BC Through 2 pts, is one line2) Line is bisector of ACn
4) AM MC3) M is midpoint of AC
Given
Def. of MidpointDef. of segment bisector
Reasons
A C
B
n M
9) AMB CMB
4) AM MC
7) AMB CMB 6) AMB& CMB are rt. s
Def. of Midpoint
Def. of lines
8) MB MB Reflexive Prop. of
5) Line is to ACn Given
Rt. Thm.
SAS10) AB BC CPCTC11) AB BC Def. of segments
Given: MB is a bisector of ACProve: ABC is isosceles
Statement1) MB is a bisector of AC Given
2) AB BC
A M
B
C
Bisector Thm.
Reasons
)
3) AB BC Def. of segmentsDef. of isosceles 4) ABC is isosceles
M is midpoint of AC
Since AB BC, BM is bisector of AC
Converse of Perpendicular Bisector Theorem
B
Find x.
If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.
5x 10 x 20 x x
4x 10 20 4x 30x 7.5
M is midpoint of AC
A 5x 10
B
CM
x 20
1010
Angle Bisector Theorem
A
D BD bisects ABC
If a point is in the interior of an angle and it lies on the angle bisector, then it is equidistant from the sides of the angle.
B C
AD DC Since AB BD, CA CD, and AB AC th AD bi t BDC b
Converse of Angle Bisector Theorem
Find m BDA if m CDB =124
B
If a point lies on the interior of an angle and isequidistant from its sides, then it lies on the anglebisector.
AB AC, then AD bisects BDC by Converse of Angle Bisector Theorem.
1m BDA m CDB2
1m BDA (124 )2
A
C
D
m BDA 62
Lesson 5-1
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
2
Write the equation of the perpendicular bisector of thesegment with endpoints C(1,-6) and B(-3,10).
Midpoint
1 2 1 2x x y yM ,
2 2
1 3 6 10M ,2 2
Slope
2 1
2 1
y ym
x x
10 6m3 1
y mx b 1y x b4
12 ( 1) b4
2 4M ,
2 2 1, 2
3 1 16m
4
4 1b 24
1b 24
94
1 9y x4 4
1m4
Write the equation of the perpendicular bisector of thesegment in point-slope form with endpoints A(-4,2) and B(6,-6).
Midpoint
1 2 1 2x x y yM ,
2 2
4 6 2 6M
Slope
2 1
2 1
y ym
x x
6 2
1 1y y m(x x )
1 15y y (x x )4
5m4
1, 2
4 6 2 6M ,2 2
2 4M ,2 2
1, 2
6 2m6 4 8m
10 4
5
1 1y y ( )45y 2 (x 1)4
5m4
Lesson 5-1
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
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