5.2 Use Perpendicular Bisectors

1. 5.2 Use Perpendicular Bisectors Hubarth Geometry

2. Perpendicular Bisector- is a segment, ray, or line that is perpendicular to a segment at its midpoint. Perpendicular Bisector theorem IF Then . . C C . . . . M M A B A B

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 . . B A . D

4. BDis the perpendicular bisector of AC. Find AD. 3x + 14 5x= x = 7 5x = 35. AD = 5(7) = Ex 1 Use the Perpendicular Bisector Theorem AD=CD Perpendicular Bisector Theorem Substitute. Solve for x.

5. In the diagram, WXis the perpendicular bisector of YZ. a. What segment lengths in the diagram are equal? b. IsV onWX ? a. WXbisects YZ , so XY =XZ. Because Wis on the perpendicular bisector of YZ, WY =WZby Theorem 5.2. The diagram shows that VY = VZ = 25. Ex 2 Use Perpendicular Bisectors b. Because VY = VZ,Vis equidistant from Yand Z. So, by the Converse of the Perpendicular Bisector Theorem, Vis on the perpendicular bisector of YZ, which is WX.

6. Concurrent- when three or more lines, rays or segments intersect at the same point. That point is called the point of concurrency. The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. The circumcenter of, an acute triangle is inside the triangle, of a right triangle is on the triangle, and of an obtuse triangle is outside the triangle.

7. In the diagram, JKis the perpendicular bisector of NL. 3. Explain why Mis on JK. NJ =LJsince JKbisects NL. NK = LK by the Perpendicular Bisector Theorem and the diagram shows ML =MN. Since ML = MN, Mis equidistant from Nand L, so by the Converse of the Perpendicular Bisector Theorem Mis on the perpendicular bisector of NLwhich is JK. Practice 1. What segment lengths are equal? Explain your reasoning. 2. Find NK. 13