1. Section 5.2
Use Perpendicular Bisectors
You’ve learned that a segment bisector intersects a segment at its ___________. A _________,
midpoint
segment
____, _____, or ________ that is ________________ to a segment at its midpoint is called a ________________ ________.
ray
line
plane
perpendicular
perpendicular
bisector

2. Section 5.2
Use Perpendicular Bisectors
A point is ______________ from two figures if the point is the _______ distance from each figure. Points on the _______________ __________ of a segment are equidistant from the segment’s ___________.
equidistant
same
perpendicular
bisector
endpoints

3. Section 5.2
Use Perpendicular Bisectors
Theorem
Theorem 5.2:
Perpendicular Bisectors Theorem
If a point is on the perpendicular bisector of a segment, it is equidistant from the segment endpoints.

4. Converse of the Perpendicular Bisectors Theorem
Section 5.2
Use Perpendicular Bisectors
Theorem
Theorem 5.3:
Converse of the Perpendicular Bisectors Theorem
If a point is equidistant from the segment endpoints, it is on the perpendicular bisector of the segment.

5. Use the Perpendicular Bisector Theorem
EXAMPLE 1
Use the Perpendicular Bisector Theorem
ALGEBRA
BD is the perpendicular bisector of AC Find AD.
AD = CD
Perpendicular Bisector Theorem
3x + 14
5x =
Substitute. 14
2x =
Subtract 3x. 7
x =
Solve for x.
AD =
5x =
5(7) =
35.

6. EXAMPLE 2
Use perpendicular bisectors
In the diagram, WX is the perpendicular bisector of YZ . a.
What segment lengths in the diagram are equal? b.
Is V on WX ? a.
XY = XZ, def. of bisector WY = WZ by Theorem 5.2 VY = VZ both = 25 b.
V is on the  bisector, WX .
VY = VZ, V is equidistant from Y and Z. (Converse of Perp. Bisector Thm.)

7. GUIDED PRACTICE
for Examples 1 and 2
In the diagram, JK is the perpendicular bisector of NL . 1.
What segment lengths are equal? Explain your reasoning.
NJ =LJ since JK bisects NL.
NK = LK by the Perp. Bisector Thm
ML = MN both = 8

8. GUIDED PRACTICE
for Examples 1 and 2
In the diagram, JK is the perpendicular bisector of NL . 2.
Find NK.
6x – 5 = 4x + 1
2x = 6
x = 3
6(3) – 5 = 13
NK = 13

9. GUIDED PRACTICE
for Examples 1 and 2
In the diagram, JK is the perpendicular bisector of NL . 3.
Explain why M is on JK .
ML = MN, M is equidistant from N and L. M is on JK by Converse of Perp. Bisector Thm .

10. Section 5.2
Use Perpendicular Bisectors
CONCURRENCY When three or more lines, segments, or rays ___________ at the same _______, they are ______________. The point of intersection is called the ____________________.
intersect
point
concurrent
point of concurrency

11. Section 5.2
Use Perpendicular Bisectors
Theorem
Theorem 5.4:
Concurrency of Perp. Bisectors of a Triangle
The perpendicular bisectors of a triangle intersect at a point which is equidistant from the vertices of the triangle.

12. EXAMPLE 3
Use the concurrency of perpendicular bisectors
FROZEN YOGURT
Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor.
Find a location for the distributor that is equidistant from the three carts.

13. EXAMPLE 3
Use the concurrency of perpendicular bisectors
Theorem use the point of concurrency of the perpendicular bisectors of the triangle formed by those points.
Use a ruler and protractor to draw the three perpendicular bisectors of ABC. Make the point of concurrency, point D, the location of the distributor.
Mark point of concurrency
Find midpoints
Draw bisector through midpoints

14. GUIDED PRACTICE
for Example 3 4.
WHAT IF?
Hot pretzels are sold from points A and B and also from a cart at point E. Where could the pretzel distributor be located if it is equidistant from those three
points? Sketch the
and show the location.
triangle
Where the perpendicular bisectors intersect –
point of concurrency

15. Section 5.2
Use Perpendicular Bisectors
CIRCUMCENTER The point of concurrency of the ____ perpendicular ___________ of a triangle is called a ______________ of the triangle. The circumcenter is _____________from the three _________ of the triangle. The prefix _________ means _________. 3
bisectors
circumcenter
equidistant
vertices
circum
around
Acute triangle Right triangle Obtuse triangle
Circumcenter, P,
INSIDE triangle.
Circumcenter, P,
ON triangle.
Circumcenter, P,
OUTSIDE triangle.