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Section 5.2Use Perpendicular
Bisectors
Vocabulary
•Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint
•Equidistant: A point that is the same distance from each figure.
•Concurrent: When three or more lines, rays, or segments intersect at the same point
•Point of Concurrency: The point of intersection of concurrent lines, rays, or segments
•Circumcenter: The point of concurrency of the three perpendicular bisectors of a triangle
Theorem 5.2: Perpendicular Bisector Theorem
In a plane, if a point is on the perpendicular
bisector of a segment, then it is equidistant from the endpoints of
the segment.
C
AP B
If CP is the bisector of AB, then CA = CB.
Theorem 5.3: Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the
endpoints of a segment, then it is on the
perpendicular bisector of the segment.
C
A
P
D
B
If DA = DB, then D lies on the bisector of AB.
Example 1Use the Perpendicular Bisector TheoremAC is the perpendicular bisector of BD. Find AD.
A
B
C
D
4x
7x-6
AD = AB4x = 7x - 6
x = 2AD = 4x = 4(2) = 8
Example 2 Use perpendicular bisectors
In the diagram, KN is the perpendicular bisector of JL.
a. What segment lengths in the diagram are equal?b. Is M on KN?a. KN bisects JL, so NJ = NL. Because K is on the perpendicular bisector of JL, KJ = KL by Thm 5.2. The diagram shows MJ = ML = 13.
K
L
13
M
13
JN
b. Because MJ = ML, M is equidistant from J and L. So, by the Converse of the Perpendicular Bisector Theorem, M is on the perpendicular of JL, which is KN.
K
L
13
M
13
JN
With a partner, do 1-2 on p. 265
checkpoints!
F
4.14.1J
Hx+1K
2xG
1. What segment lengths are equal?
2. Find GH.
In the diagram, JK is the perpendicular bisector of GH.
Theorem 5.4 Concurrency of Perpendicular Bisectors of a Triangle
The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
If PD, PE, and PF are perpendicular bisectors, then PA = PB = PC.
B
E
CF
D
A
P
Example 3Use the concurrency of perpendicular bisectors
The perpendicular bisectors of ΔMNO meet at point S. Find SN.
Using Theorem 5.4, you know that point S is equidistant from the
vertices of the triangle. So, SM = SN = SO.
SN = SMSN = 9
N
Q
OR
S
29
M
P
Do #3 on p. 265
checkpoint!
B
E
C
F
G
6
15
A
D
12
The perpendicular bisectors of ΔABC meet at point G. Find GC.
Essential QuestionWhat do you know
about perpendicular bisectors?
