Section 5.2 Use Perpendicular Bisectors. Vocabulary Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

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Section 5.2 Use 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 A P 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 1 U se the Perpendicular Bisector Theorem AC is the perpendicular bisector of BD. Find AD. A B C D 4x 7x-6 AD = AB 4x = 7x - 6 x = 2 AD = 4x = 4(2) = 8

Example 2 U se 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 J N

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 J N

With a partner, do 1-2 on p. 265

checkpoints! F 4.1 J H x+1 K 2x G 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 C F D A P

Example 3 U se 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 = SM SN = 9 N Q O R S 2 9 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 Question What do you know about perpendicular bisectors?