5.3 Use Angle Bisectors of Triangles Hubarth Geometry.

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5.3 Use Angle Bisectors of Triangles Hubarth Geometry

Angle Bisector Theorem 1 2 If then A. B 1 2. B C A D

Find the measure of  GFJ. Ex 1 Use the Angle Bisector Theorem

BP = CP Set segment lengths equal. x + 3 = 2x –1 Substitute expressions for segment lengths. 4 = x Solve for x. Point P lies on the bisector of A when x = 4. For what value of x does P lie on the bisector of A ? Ex 2 Use Algebra to Solve a Problem

Concurrency of Angle Bisector of a Triangle The angle bisector of a triangle intersects at a point that is equidistant from the sides of a triangle. B E C A D F

In the diagram, N is the incenter of ABC. Find ND. By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter N is equidistant from the sides of ABC. So, to find ND, you can find NF in NAF. Use the Pythagorean Theorem. Ex 3 Use the Concurrency of Angle Bisectors c = 2 a + b 2 2 Pythagorean Theorem NF = 2 Substitute known values. 400 =NF Multiply. 144 =NF 2 12 =NF Subtract 256 from each side. Take the positive square root of each side. Because NF = ND, ND = 12.

Practice In Exercises 1–3, find the value of x. 1. A B C P 15 A B C P A B C P 5 4. Do you have enough information to conclude that QS bisects PQR ? Explain. 5. In diagram, suppose you are not given AF or AN, but you are given that BF = 12 and BN = 13. Find ND. 5