1. Vocabulary YOU NEED TO TAKE NOTES ON THIS!

2. Concept

3. Example 1 Use the Perpendicular Bisector Theorems A. Find the measure of BC. Answer: 8.5

4. Example 1 Use the Perpendicular Bisector Theorems B. Find the measure of XY. Answer: 6

5. Example 1 Use the Perpendicular Bisector Theorems C. Find the measure of PQ. Answer: 7

6. A.A B.B C.C D.D Example 1 A.4.6 B.9.2 C.18.4 D.36.8 A. Find the measure of NO.

7. A.A B.B C.C D.D Example 1 A.2 B.4 C.8 D.16 B. Find the measure of TU.

8. A.A B.B C.C D.D Example 1 A.8 B.12 C.16 D.20 C. Find the measure of EH.

9. Concept

11. Special case - right triangles In the special case of a right triangle, the circumcenter (C in the figure at right) lies exactly at the midpoint of the hypotenuse (longest side)

12. Example 2 Use the Circumcenter Theorem GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points.

14. Example 2 Use the Circumcenter Theorem Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle. Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle. C

15. A.A B.B Example 2 A.No, the circumcenter of an acute triangle is found in the exterior of the triangle. B.Yes, circumcenter of an acute triangle is found in the interior of the triangle. BILLIARDS A triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle?

16. Concept Distance is always measured along a perpendicular line

17. Example 3 Use the Angle Bisector Theorems A. Find DB. Answer: DB = 5 DB= DCAngle Bisector Theorem DB= 5Substitution

18. Example 3 Use the Angle Bisector Theorems B. Find  WYZ. Answer: m  WYZ = 28

19. Example 3 Use the Angle Bisector Theorems C. Find QS. Answer: So, QS = 4(3) – 1 or 11.

20. A.A B.B C.C D.D Example 3 A.22 B.5.5 C.11 D.2.25 A. Find the measure of SR.

21. A.A B.B C.C D.D Example 3 A.28 B.30 C.15 D.30 B. Find the measure of  HFI.

22. A.A B.B C.C D.D Example 3 A.7 B.14 C.19 D.25 C. Find the measure of UV.

23. Concept

24. Example 4 Use the Incenter Theorem A. Find SU if S is the incenter of ΔMNP. Answer: SU = 6

25. Example 4 Use the Incenter Theorem Answer: SU = 6

26. Example 4 Use the Incenter Theorem B. Find  SPU if S is the incenter of ΔMNP. Answer: m  SPU = (62) or 31 __ 1 2

27. A.A B.B C.C D.D Example 4 A.12 B.144 C.8 D.65 A. Find the measure of GF if D is the incenter of ΔACF.

28. A.A B.B C.C D.D Example 4 A.58° B.116° C.52° D.26° B. Find the measure of  BCD if D is the incenter of ΔACF.