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Chapter 5 Properties of Triangles Perpendicular and Angle Bisectors Sec 5.1 Goal: To use properties of perpendicular bisectors and angle bisectors Perpendicular Bisector Perpendicular Bisector – a segment, ray, line, or plane that is perpendicular to a segment at its midpoint. C A M P If CP is a perpendicular bisector of AB, then AP PB and CP AB B Equidistant Equidistant from two points means that the distance from each point is the same. C A C is equidistant from A and B; therefore, CA = CB B Perpendicular Bisector Theorem Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. C A If CD is the perpendicular bisector of AB, then CA CB. Likewise, DA DB. CD AB P D B Converse of the Perpendicular Bisector Theorem Converse of the Perpendicular Bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment. C A P B D If DA DB, then D lies on the perpendicular bisector of AB. Example Does D lie on the perpendicular bisector of AB ? Why ? C A B 7 6 D Example Is CD a perpendicular bisector of AB ? Why, or why not? C A D B Distance from a point to a line The distance from a point to a line is defined to be the shortest distance from the point to the line. This distance is the length of the perpendicular segment. Q m P The distance from point P to line m is the length of QP Angle Bisector Theorem Angle Bisector Theorem – If a point (D) is on the bisector of an angle, then it is equidistant from the two sides of the angle. B If AD, is the angle bisector of BAC, then BD CD A C BD AB and DC AC D Converse of the Angle Bisector Theorem If DB DC, then AD lies on the bisector of BAC B A D C Converse of the Angle Bisector Theorem – If a point is on the interior of an angle, and is equidistant from the sides of the angle, then it lies on the bisector of the angle. Examples Does the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB? A B A D C P B A C C P B Examples Does the information given in the diagram allow you to conclude that P is on the angle bisector of angle A? P 6 P 6 P Examples Draw the segment that represents the distance indicated. L R T M R perpendicular to LM R P A T perpendicular to AP Examples Name the segment whose length represents the distance between: M to AT M A T to HM H to MT H T