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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Presentation transcript:

Section 5.2 Use Perpendicular Bisectors

Vocabulary Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Equidistant: A point is equidistant from two figures if the point is the same distance from each figure. Concurrent: When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments.

Point of Concurrency: The point of intersection of concurrent lines, rays, or segments is called the point of concurrency. Circumcenter: The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the 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

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. The point is called the circumcenter of the triangle. If PD, PE, and PF are perpendicular bisectors, then PA = PB = PC. Point P is the circumcenter. 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. You know that point S (the circumcenter) 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

The intersection point of the perpendicular bisectors, the circumcenter, is the center of the circumscribed circle of a triangle.