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EXAMPLE 4

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 stated on page 18. EXAMPLE 4.

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EXAMPLE 4

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  1. In the diagram, Nis the incenter of ABC. Find ND. By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter Nis equidistant from the sides of ABC. So, to find ND, you can find NFin NAF. Use the Pythagorean Theorem stated on page 18. EXAMPLE 4 Use the concurrency of angle bisectors SOLUTION

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

  3. By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter Nis equidistant from the sides of ABC. So, to find ND, you can find NFin FNB. Use the Pythagorean Theorem stated on page 18. for Example 4 GUIDED PRACTICE WHAT IF? 5. In Example 4, suppose you are not given AF or AN, but you are given that BF = 12 and BN = 13. Find ND. SOLUTION

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

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