3.6—Bisectors of a Triangle Warm Up 1. Draw a triangle and construct the bisector of one angle. 2....

3.6—Bisectors of a Triangle

Warm Up

1. Draw a triangle and construct the bisector of one angle.

2. JK is perpendicular to ML at its midpoint K. List the congruent segments. Draw a picture.

3.6—Bisectors of a Triangle

Objective: Use properties of _____________ bisectors and ___________ bisectors of a __________.

perpendicularangle triangle

Concurrent lines: three or more lines that intersect in the _______ ________

Point of concurrency: the _________ of ________________ of concurrent lines

same point

pointintersection

The circumcenter can be: inside the triangle (acute triangles), on the triangle (right triangles), or outside the triangle (obtuse triangles). 

_______________ Bisector of a triangle: a ________ that is perpendicular to a side of the triangle at the __________ of the side.

________________ of a triangle: the point of ____________________ of the perpendicular _______________ of a triangle.

Perpendicularline

midpoint

Circumcenterconcurrency

bisectors

Examples: 1. The perpendicular bisectors of ΔHIJ 2. R is the circumcenter of ΔOPQ, meet at K, IJ = 18, and KJ = 12. OS = 10, QR = 12, and PQ = 22.

a. Find HK. a. Find OP.

b. Find IM. b. Find RP.

c. Find KM. c. Find RT.

________ _____________ of a triangle: a bisector of an _________ of a triangle

_______________ of a triangle:the point of __________________ of the angle bisectors

Angle Bisector angle

Incenterconcurrency

The incenter is always inside the triangle.

Examples: 3. D is the incenter of ΔABC, 4. The angle bisectors of ΔABC meet at P,

a. Find DE. PR = 3 and PC = 5.

b. Find GC. a. Find QP.

b. Find QC.

Construction – Perpendicular to a line from a point not on the line.

Construct a perpendicular to the given line and through the given point.

. A.

B. . 

Construct the Circumcenter and the Circumscribed Circle for the triangle.

Construct the Incenter and the Inscribed Circle for the triangle.