3. Find each measure of MN. Justify
Perpendicular Bisector Theorem

4. Write an equation to solve for a.
Justify
3a + 20 = 2a + 26
Converse of  Bisector Theorem

5. Find the measures of BD and
BC. Justify
BD = 12
BC =24
Converse of  Bisector Theorem

6. Find the measure of BC.
Justify
BC = 7.2
 Bisector Theorem

7. Write the equation to solve for x. Justify your equation.
3x + 9 = 7x – 17
 Bisector Theorem

8. Find the measure.
mEFH, given that mEFG = 50°.
Justify
m EFH = 25
Converse of the  Bisector Theorem

9. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1).

10. Perpendicular Bisectors
of a triangle…
bisect each side at a right angle
meet at a point called the circumcenter
The circumcenter is equidistant from the 3 vertices of the triangle.
The circumcenter is the center of the circle that is circumscribed about the triangle.
The circumcenter could be located inside, outside, or ON the triangle. C

11. Angle Bisectors Paste-able! of a triangle… bisect each angle
meet at the incenter
The incenter is equidistant from the 3 sides of the triangle.
The incenter is the center of the circle that is inscribed in the triangle.
The incenter is always inside the circle. I

12. DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.

13. MZ and LZ are perpendicular bisectors of ∆GHJ. Find GM

14. Z is the circumcenter of ∆GHJ. GK and JZ
GK = JZ = 19.9

15. Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).

16. MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.

17. MP and LP are angle bisectors of ∆LMN. Find mPMN.

18. 5-3: Medians and AltitudesZ Y X C B 
Medians of triangles:
Endpoints are a vertex
and midpoint of opposite side.
Intersect at a point called the centroid
Its coordinates are the average of the 3 vertices.
The centroid is ⅔ of the distance from each vertex to the midpoint of the opposite side.
The centroid is always located inside the triangle. P

19. 5-3: Medians and Altitudes
Altitudes of a triangle:
A perpendicular segment from a vertex to the line containing the opposite side.
Intersect at a point called the orthocenter.
An altitude can be inside, outside, or on the triangle.

20. In ∆LMN, RL = 21 and SQ =4. Find LS.

21. In ∆LMN, RL = 21 and SQ =4. Find NQ.

22. In ∆JKL, ZW = 7, and LX = 8.1. Find KW.

23. Example 2: Problem-Solving Application
A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance?

24. Find the average of the x-coordinates and the average of the y-coordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle.

25. Find the orthocenter of ∆XYZ with vertices X(3, –2), Y(3, 6), and Z(7, 1).