Multiple choice

A circle is inscribed in triangle ABC, touching all three sides. The lengths of the sides are AB = 13 cm, BC = 14 cm, and AC = 15 cm. Find the radius of the incircle.

  1. 4

  2. 2

  3. 12

  4. 13

  5. 5
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A Correct answer
Explanation

Area of triangle = sqrt(s(s-a)(s-b)(s-c)) where s = (13+14+15)/2 = 21. Area = sqrt(21 * 8 * 7 * 6) = sqrt(7056) = 84. Inradius r = Area / s = 84 / 21 = 4.

AI explanation

First, find the semi-perimeter of the triangle using s = (13 + 14 + 15) / 2, which gives s = 21 cm. Next, calculate the area of triangle ABC using Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) = sqrt(21 * 8 * 7 * 6) = sqrt(7056) = 84 cm^2. Finally, use the incircle radius formula r = Area / s, resulting in r = 84 / 21 = 4 cm.

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