AMSCO'S Geometry. New York - Rye High School

Perpendicular Bisector of a Line Segment 193Theorem 5.3bIf a point is on the perpendicular bisector of a line segment, then it is equidistantfrom the endpoints of the line segment.Given Point P on the perpendicular bisector of AB.Prove PA = PBCPProof Let M be the midpoint of AB. Then AM > BMand PM > PM. Perpendicular lines intersect toform right angles, so PMA PMB. By SAS,PMA PMB. Since corresponding parts ofcongruent triangles are congruent, PA > PB andPA PB.AMDBTheorems 5.3a and 5.3b can be written as a biconditional.Theorem 5.3A point is on the perpendicular bisector of a line segment if and only if it isequidistant from the endpoints of the line segment.Methods of Proving Linesor Line Segments PerpendicularTo prove that two intersecting lines or line segments are perpendicular, provethat one of the following statements is true:1. The two lines form right angles at their point of intersection.2. The two lines form congruent adjacent angles at their point of intersection.3. Each of two points on one line is equidistant from the endpoints of a segmentof the other.Intersection of the Perpendicular Bisectors ofthe Sides of a TriangleWhen we draw the three perpendicular bisectorsof the sides of a triangle, it appears that the threelines are concurrent, that is, they intersect in onepoint.Theorems 5.3a and 5.3b allow us to prove thefollowing perpendicular bisector concurrencetheorem.ANMPCLB