1. Section 5.2 Notes

2. Perpendicular Bisector
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. A point is equidistant from two figures if the point is the same distance from each figure.

3. 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.

4. 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.

5. Example 1
a. Line SR is the perpendicular bisector of seg. PQ. Find PR.  

6. b. Line AC is the perpendicular bisector of seg. BD. Find AD.

7. Example 2
a. In the diagram, line JM is the perpendicular bisector of seg. HK. Which lengths in the diagram are equal? Is L on line JM?
Yes, by the Converse of the Perpendicular Bisector Theorem.

8. b. In the diagram, line KN is the perpendicular bisector of seg. JL
b. In the diagram, line KN is the perpendicular bisector of seg. JL. What segment lengths in the diagram are equal? Is M on line KN?
Yes, by the Converse of the Perpendicular Bisector Theorem.

9. Concurrency
When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments. The point of the intersection of the lines, rays, or segments is called the point of concurrency.

10. 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.

11. Example 3
a. Each of three forest ranger stations is the same distance from the main office. Describe how to find the location of the office. Draw the perpendicular bisectors of the three sides. Those three lines meet at a point, and that point is the location of the office.

12. b. Three friends are playing catch
b. Three friends are playing catch. You want to join and position yourself so that you are the same distance from your friends. Find a location for you to stand. Connect points A, B, and C. Then use a ruler and protractor to draw the three perpendicular bisectors of ΔABC. The point of concurrency D is a location for you to stand.

13. c. 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 triangle and show the location.

14. The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. This point is the center of a circle that passes through all three vertices.

15. As shown above, the location of P depends on the type of triangle
As shown above, the location of P depends on the type of triangle. The circle with point P as its center is said to be circumscribed about the triangle.