1. Bisectors
Concept 35

2. Perpendicular bisector – a segment, line or plane that intersects a segment at its midpoint and creates right angles.
Vocabulary

3. Concept

4. BC = AC Perpendicular Bisector Theorem BC = 8.5 Substitution
Use the Perpendicular Bisector Theorems
A. Find BC.
BC = AC Perpendicular Bisector Theorem
BC = 8.5 Substitution
Answer: 8.5
Example 1

5. B. Find XY. Answer: 6 Use the Perpendicular Bisector Theorems
Example 1

6. PQ = RQ Perpendicular Bisector Theorem 3x + 1 = 5x – 3 Substitution
Use the Perpendicular Bisector Theorems
C. Find PQ.
PQ = RQ Perpendicular Bisector Theorem
3x + 1 = 5x – 3 Substitution
1 = 2x – 3 Subtract 3x from each side.
4 = 2x Add 3 to each side.
2 = x Divide each side by 2.
So, PQ = 3(2) + 1 = 7.
Answer: 7
Example 1

7. A. Find NO.
A. 4.6
B. 9.2
C. 18.4
D. 36.8
Example 1

8. B. Find TU.
A. 2
B. 4
C. 8
D. 16
Example 1

9. C. Find EH.
A. 8
B. 12
C. 16
D. 20
Example 1

10. Angle bisector – a line bisects the angle of a triangle and divides it into 2 equal angles.
Vocabulary

11. Concept

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

13. WYZ  XYW Definition of angle bisector
Use the Angle Bisector Theorems
B. Find mWYZ.
WYZ  XYW Definition of angle bisector
mWYZ = mXYW Definition of congruent angles
mWYZ = 28 Substitution
Example 3

14. QS = SR Angle Bisector Theorem 4x – 1 = 3x + 2 Substitution
Use the Angle Bisector Theorems
C. Find QS.
QS = SR Angle Bisector Theorem
4x – 1 = 3x + 2 Substitution
x – 1 = 2 Subtract 3x from each side.
x = 3 Add 1 to each side.
Answer: So, QS = 4(3) – 1 or 11.
Example 3

15. Angle Bisectors of a triangle
Angle bisectors do have a vertex as an endpoint.

16. Perpendicular bisector – a segment, line or plane that intersects a segment at its midpoint and creates right angles.
Vocabulary

17. Perpendicular Bisectors of Triangles
Perpendicular bisectors do NOT have a vertex as an endpoint (no endpoints)
Circumcenter is point where perpendicular bisector meet at one point.
Where is circumcenter located?
Acute Triangle
Inside triangle
Right Triangle
On the midpoint of the hypotenuse
Obtuse Triangle
Outside the longest side of the triangle

19. Perpendicular Bisector Theorem
DA = BD = CD
Concept

20. Example 1
The perpendicular bisectors of
Find PX

21. Example 2
The perpendicular bisectors of
meet at point G. Find GA.

22. 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.
Example 2

23. Use the Circumcenter Theorem
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
Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.
Example 2

24. Angle Bisectors of a triangle
Angle bisectors do have a vertex as an endpoint.
The Incenter is the point at which angle bisector of a triangle meet at one point.
Where is the incenter located?
Always inside the triangle for any type of triangle.

25. Angle Bisector Theorem
Concept

26. Example 1
The angle bisector of
ABC meet at point G.
m DAG = ___________ FG = ______ = ______
m ECG = ___________ m
GBF = __________
GD = ___________
AF = ___________

27. Example 2
The angle bisectors of XYZ meet at point P.
Find PM.
Find PY.

28. Find mSPU if S is the incenter of ΔMNP.
Use the Incenter Theorem
Find mSPU if S is the incenter of ΔMNP.
Find SR
Example 4

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

30. B. Find the measure of BCD if D is the incenter of ΔACF.
Example 4