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A circle is inscribed in a right triangle ABC with the right angle at C. The circle touches the hypotenuse AB at point D, where AD = 7 and DB = 13. Find the area of triangle ABC.

1. 91
2. 96
3. 100
4. 104

Correct answer: 91

Solution

By equal tangent lengths: tangent from A = 7 (to AB and to AC), tangent from B = 13, tangent from C = r (inradius). So AC = 7+r, BC = 13+r, AB = 20. Pythagorean theorem: (7+r)² + (13+r)² = 400 gives r² + 20r = 91. Area = r*(20+r) = 20r + r² = 91.

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