CK-12 Geometry Concepts

5.2. Perpendicular Bisectors http://www.ck12.org

2. Construct the perpendicular bisector for all three sides.

The three perpendicular bisectors all intersect at the same point, called the circumcenter.

Circumcenter:The point of concurrency for the perpendicular bisectors of the sides of a triangle.

3. Erase the arc marks to leave only the perpendicular bisectors. Put the pointer of your compass on the circumcenter.
   Open the compass so that the pencil is on one of the vertices. Draw a circle. What happens?

The circumcenter is the center of a circle that passes through all the vertices of the triangle. We say that this circle
circumscribesthe triangle. This means thatthe circumcenter is equidistant to the vertices.

Concurrency of Perpendicular Bisectors Theorem:The perpendicular bisectors of the sides of a triangle intersect
in a point that is equidistant from the vertices.

IfPC,QC, andRCare perpendicular bisectors, thenLC=MC=OC.

Example A

Findxand the length of each segment.

From the markings, we know that

←→

W Xis the perpendicular bisector ofXY. Therefore, we can use the Perpendicular
Bisector Theorem to conclude thatW Z=WY. Write an equation.

2 x+ 11 = 4 x− 5

16 = 2 x

8 =x

To find the length ofW ZandWY, substitute 8 into either expression, 2( 8 )+ 11 = 16 + 11 =27.

Example B

←→
OQis the perpendicular bisector ofMP.

a) Which segments are equal?

b) Findx.

c) IsLon

←→

OQ? How do you know?

Answer:

a)ML=LPbecause they are both 15.

MO=OPbecauseOis the midpoint ofMP

MQ=QPbecauseQis on the perpendicular bisector ofMP.

b)

4 x+ 3 = 11

4 x= 8

x= 2

c) Yes,Lis on

←→

OQbecauseML=LP(Perpendicular Bisector Theorem Converse).