November 27, bisectors of triangles ink.notebook. Page 168. Page 167. Page 166. Ch 5 Triangle Relationships. 5.1 Bisectors of Triangles

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1 5.1 bisectors of triangles ink.notebook Page 166 Page 167 Page 168 Ch 5 Triangle Relationships 5.1 Bisectors of Triangles Page 169 Page 170 Page 171 1

2 Lesson Objectives Standards Lesson Notes Lesson Objectives Standards Lesson Notes 5.1 Bisectors of Triangles After this lesson, you should be able to successfully use perpendicular bisectors to solve problems. Press the tabs to view details. Press the tabs to view details. Lesson Objectives Standards Lesson Notes G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.10 Prove theorems about triangles

3 ***In Geometry, distance means the shortest length between two objects.*** Perpendicular Bisector a segment, ray, line, or plane that is perpendicular to a segment at its midpoint Concurrent when 3 or more lines, rays, or segments intersect in the same point Equidistant the same distance away from 2 figures. Point of Concurrency Point of Concurrency The point of intersection of concurrent lines, rays, or segments. 3

4 5.1 bisectors of triangles ink.notebook Circumcenter The point of concurrency of the perpendicular bisectors of a 3 triangle. circumcenter Perpendicular Bisector Theorem perpendicular bisector In a plane, if a point is on the equidistant from the of a segment, then it is endpoints of the segment. C A P B CB 4

5 5.1 bisectors of triangles ink.notebook Converse of the Perpendicular Bisector Thm In a plane, if a point is equidistant from the endpoints perpendicular of a segment, then it is on the bisector of the segment. C A If DA = DB, then D lies on the B P bisector D Concurrency of Perpendicular Bisectors of a Triangle CIRCUMCENTER THM The perpendicular bisectors of a triangle intersect at a equidistant vertices of from the point that is the triangle. B D P E perpendicular bisectors then PC = PB PA = A F C 5

6 1. Find the measure of FM. FM = x = Find AD. 7x - 6 B A 4x C D x = AD = 6

7 5.1 bisectors of triangles ink.notebook Point P is the circumcenter of EMK. List any segment(s) congruent to each segment below. E Y M R N K P Angle Bisectors r to isec le b ang Incenter Another special segment, ray, or line is an angle bisector, which divides an angle into 2 congruent angles. The point of concurrency of the 3 angle bisectors of a triangle. incenter 7

8 Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle,then it lies on the bisector of the angle. Then BAD CAD Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If AD bisects BAC and DB AD and DC AC then DB =. DC INCENTER THEOREM The angle bisector of a triangle intersect at a point that is equidistant from the sides of the triangle. A If AP, BP, and CP are angle bisectors of ΔABC then PD = PE =. PF D P F B E C 8. Find the measure of CBE. 8

9 5.1 bisectors of triangles ink.notebook Point U is the incenter of GHY. Find each measure below. 10. m GHY = 11. m UGM = 12. m PHU = 13. m HYG = 9

10 Find each measure. On the Worksheet 1. m ABE = 2. MK = Find each measure. Find the value of x

11 4. For what value of x does P lie on the bisector of J? PRACTICE Find each measure. 1. FG = 2. KL = Find each measure. 3. TU = 4. m LYFG = 11

12 Find each measure. 5. IU = 6. m MYW = Point P is the circumcenter of EMK. List any segment(s) congruent to each segment below. Point A is the incenter of PQR. Find each measure below. 10. m ARU = 11. x = m QPR = Point A is the incenter of PQR. Find each measure below. 13. m YLA = 14. m LGA = 12. m PQU = 12

13 15. A triangular entranceway has walls with corner angles of 50, 70, and 60. The designer wants to place a tall bronze sculpture on a round pedestal in a central location equidistant from the three walls. How can the designer find where to place the sculpture? 16. Joanna has a flat wooden triangular piece as part of a wind chime. The piece is suspended by a wire anchored at a point equidistant from the sides of the triangle. Where is the anchor point located? 17. Marsha and Bill are going to the park for a picnic. The park is triangular. One side of the park is bordered by a river and the other two sides are bordered by busy streets. Marsha and Bill want to find a spot that is equally far away from the river and the streets. At what point in the park should they set up their picnic? 18. Martin has 3 grown children. The figure shows the locations of Martin's children on a map that has a coordinate plane on it. Martin would like to move to a location that is the same distance from all three of his children. What are the coordinates of the location on the map that is equidistant from y all three children? x 13

14 Answers: 14