1. Bisectors in Triangles

2.  Since a triangle has ________ sides, it has three ___________ ____________ The perpendicular bisector of a side of a _____________ does not always pass through the _____________ ________ 3 perpendicular bisectors triangle opposite vertex

3.  When three or more lines intersect at one point, the lines are said to be _________________.  The ___________ ____ ___________ is the point where these lines intersect.  The three perpendicular bisectors of a triangle are ______________. This point of concurrency is the ______________ of the ____________ concurrent pointof concurrency concurrent circumcenter triangle

4.  The circumcenter of ΔABC is the center of its _____________________ circle. circumscribed

5.  The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

6. DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC. G is the _______________ of ∆ABC. G is equidistant from the vertices of ∆ABC --using the circumcenter theorem GB = GA = GC = 13.4 circumcenter

7. MZ is a perpendicular bisector of ∆GHJ. Find the measures. ZJ = ZH = GM = KG = 19.9 14.5 18.6

8. Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6). H K J Step 1---Graph the triangle

9. Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6). H K J Step 2 – graph the midpoints of each side

10. Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6). H K J Step 3 – find the slope of each side, then find the perpendicular slope of each side m HJ = m HK = m KJ = und zero m HK = m HJ = m KJ = zero und (5, 3) is the circumcenter of the triangle

11. Find the circumcenter of ∆GOH with vertices G(0, –4), O(0, 0), and H(6, 0).

12. A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle. The distance between a point and a line is the length of the perpendicular segment from the point to the line. Remember!

13. Unlike the circumcenter, the incenter is always inside the triangle. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.

14. MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN. P is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN. The distance from P to LM is 5. So the distance from P to MN = ________. 5

15. MP and LP are angle bisectors of ∆LMN. Find m  PMN. m  PLN = 50 o m  MLN = 100° m  L + m  N + m  M = 180° 100 + 20 + m  M= 180 m  M = 60° m  PMN= 30 o