1. Section 5.2 Perpendicular Bisectors Chapter 5 PropertiesofTriangles

2. a segment, ray, line, or plane that is perpendicular to a segment at its midpoint. A B C P P is the midpoint CP is the perpendicular bisector of AB Perpendicular Bisector

3. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. A B C P X If CP is the perpendicular bisector of AB, then CA = CB XA = XB

4. 6)  ADC   BDC 3)  1 and  2 are right angles. 1) Given 2) Def. of Bisector 3) Def. of  5) Reflexive 6) SAS 7) CPCTC 4)  1   2 4) Def. of right  ’s 8) AC  BC 8) Def. of  1) CD  bisector of AB 2) AD  DB 5) CD  CD 7) AC  BC Given: CD is  bisector of AB Prove: AC  BC C A D B 1 2

5. Perpendicular Bisector Converse If a point is equidistant from the endpoints of the segment, then it lies on the perpendicular bisector of the segment. If XA = XB, then X lies on the perpendicular bisector of AB. A B P X

6. The distance from a point and a line is the length of the perpendicular segment from the point to the line. Distance from a Point to a Line NO YES

7. MN is the perpendicular bisector of ST. What segments are equal ? MT = MS N Q T S M 12

8. MN is the perpendicular bisector of ST. Why is Q on MN ? Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. N Q T S M 12

9. Given: AB is the  bisector of CD. Find each angle or segment measure. 1)m  1 = 2)m  2 = 3)m  4 = 4)m  3 = 5)BD = 6)AD = 7)BE =

10. 1) Fold the triangle to form the perpendicular bisectors of the sides. Do the three bisectors intersect at the same point? 2) Label the point of intersection P. Measure AP, BP and CP. What do you observe?

11. Look at the following pictures. AB is a Perpendicular Bisector in each of the following triangles A B A B A B A B Note: The perpendicular bisector does not necessarily go through a vertex of the triangle.

12. A B C A B C Step 1: Find the midpoint of the side. Step 2: Draw a line  from the midpoint. Draw the perpendicular bisector of BC in the following triangles.

13. Three or more lines that intersect at the same point. The point of intersection of the lines is called the

14. Draw the perpendicular bisector of all the sides of the triangle. First, find the midpoints of each side of the triangle. Next, draw a line perpendicular to the side from each midpoint. The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle.

15. The circumcenter is the center of the circumscribed circle that passes through the vertices of the triangle.

16. The circumcenter is inside the triangle if the triangle is acute.

17. Where would the circumcenter be in an obtuse triangle?

18. Where would the circumcenter be in a right triangle?

19. Determine if the circumcenters of the following triangles will be inside, outside, or on the triangle.

23. What kind of triangle would have at least one perpendicular bisector go through a vertex?

24. What kind of triangle would have all three perpendicular bisectors go through the vertices?

25. Theorem 5.4: Concurrency of Perpendicular Bisectors of a Triangle: The  bisectors of a  intersect at a point (the circumcenter) that is equidistant from the vertices of the triangle. A C B P PA = PC = PB What segments would be equal? What are these segments called in relationship to the circle? RADIUS

26. B C D E F A G 4 7 6 The circumcenter of  ACE is at G. Find GA and AB. GC = GA = GE Therefore GA = 7 AB = BC Therefore AB = 6