1. EXAMPLE 4 Use the concurrency of angle bisectors In the diagram, N is the incenter of ABC. Find ND. SOLUTION 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 stated on page 18.

2. EXAMPLE 4 Use the concurrency of angle bisectors c = 2 a + b 22 Pythagorean Theorem NF + 16 2 2 20 = 2 Substitute known values. 400 =NF + 256 2 Multiply. 144 =NF 2 12 =NF Subtract 256 from each side. Take the positive square root of each side. Because NF = ND, ND = 12.

3. GUIDED PRACTICE for Example 4 5. WHAT IF? In Example 4, suppose you are not given AF or AN, but you are given that BF = 12 and BN = 13. Find ND. SOLUTION 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 FNB. Use the Pythagorean Theorem stated on page 18.

4. GUIDED PRACTICE for Example 4 c = 2 a + b 22 Pythagorean Theorem NF + 12 2 2 13 = 2 Substitute known values. 169 =NF + 144 2 Multiply. 25 =NF 2 5 =NF Subtract 256 from each side. Take the positive square root of each side. Because NF = ND, ND = 5.