5.2 Use Perpendicular Bisectors

Transcription

1 5.2 Use Perpendicular isectors Goal p Use perpendicular bisectors to solve problems. Your Notes VOULRY Perpendicular bisector Equidistant oncurrent Point of concurrency ircumcenter THEOREM 5.2: PERPENIULR ISETOR THEOREM In a plane, if a point is on the perpendicular bisector of a segment, then it is P from the endpoints of the segment. P is the bisector of }, then 5. THEOREM 5.3: ONVERSE O THE PERPENIULR ISETOR THEOREM In a plane, if a point is equidistant from the endpoints of a segment, P then it is on the of the segment. If 5, then lies on the of }. 126 Lesson 5.2 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

2 5.2 Use Perpendicular isectors Goal p Use perpendicular bisectors to solve problems. Your Notes VOULRY Perpendicular bisector segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Equidistant point is equidistant from two figures if the point is the same distance from each figure. oncurrent 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. ircumcenter The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. THEOREM 5.2: PERPENIULR ISETOR THEOREM In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant P from the endpoints of the segment. P is the bisector of }, then 5. THEOREM 5.3: ONVERSE O THE PERPENIULR ISETOR THEOREM In a plane, if a point is equidistant from the endpoints of a segment, P then it is on the perpendicular bisector of the segment. If 5, then lies on the bisector of }. 126 Lesson 5.2 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

3 Example is the perpendicular bisector of }. ind. 5 Use the Perpendicular isector Theorem 5 Substitute. x 5 Solve for x x 2 6 4x Perpendicular isector Theorem Example 2 Use perpendicular bisectors In the N is the perpendicular bisector of } JL. a. What segment lengths in the diagram are equal? b. Is M N? J N L M N bisects } JL, so 5. ecause is on the perpendicular bisector of } JL, 5 by Theorem 5.2. The diagram shows that 5 5. b. ecause MJ 5 ML, M is from J and L. So, by the, M is on the perpendicular bisector of } JL, which N. heckpoint In the J is the perpendicular bisector of } GH. 1. What segment lengths are equal? 2. ind GH. 4.1 J 4.1 G 2x x 1 1 H opyright Holt Mcougal. ll rights reserved. Lesson 5.2 Geometry Notetaking Guide 127

4 Example is the perpendicular bisector of }. ind. 5 4x 5 7x 2 6 Use the Perpendicular isector Theorem x 5 2 Solve for x. 5 4x 5 4(2) x 2 6 4x Perpendicular isector Theorem Substitute. Example 2 Use perpendicular bisectors In the N is the perpendicular bisector of } JL. a. What segment lengths in the diagram are equal? b. Is M N? J N L M N bisects } JL, so NJ 5 NL. ecause is on the perpendicular bisector of } JL, J 5 L by Theorem 5.2. The diagram shows that MJ 5 ML 5. b. ecause MJ 5 ML, M is equidistant from J and L. So, by the onverse of the Perpendicular isector Theorem, M is on the perpendicular bisector of } JL, which N. heckpoint In the J is the perpendicular bisector of } GH. 1. What segment lengths are equal? G 5 H, JG 5 JH, G 5 H 2. ind GH. 4.1 J 4.1 GH 5 4 G 2x x 1 1 H opyright Holt Mcougal. ll rights reserved. Lesson 5.2 Geometry Notetaking Guide 127

5 THEOREM 5.4: ONURRENY O PERPENIULR ISETORS O TRINGLE The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices E of the triangle. P If } P, } PE, and } P are perpendicular bisectors, then P 5 5. Example 3 Use the concurrency of perpendicular bisectors ootball Three friends are playing catch. You want to join and position yourself so that you are the same distance from your friends. ind a Theorem 5.4 shows you that you can find a point equidistant from three points by using the of the triangle formed by those points. opy the positions of points,, and and connect those points to draw n. Then use a ruler and a protractor to draw the three of n. The point of concurrency is a heckpoint omplete the following exercise. Homework 3. In Example 3, your friend at location wants to move to a location that is the same distance from everyone else. ind a new location for. 128 Lesson 5.2 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

6 THEOREM 5.4: ONURRENY O PERPENIULR ISETORS O TRINGLE The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices E of the triangle. P If } P, } PE, and } P are perpendicular bisectors, then P 5 P 5 P. Example 3 Use the concurrency of perpendicular bisectors ootball Three friends are playing catch. You want to join and position yourself so that you are the same distance from your friends. ind a Theorem 5.4 shows you that you can find a point equidistant from three points by using the perpendicular bisectors of the triangle formed by those points. opy the positions of points,, and and connect those points to draw n. Then use a ruler and a protractor to draw the three perpendicular bisectors of n. The point of concurrency is a heckpoint omplete the following exercise. Homework 3. In Example 3, your friend at location wants to move to a location that is the same distance from everyone else. ind a new location for. 128 Lesson 5.2 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.