Right triangle with hypotenuse has an incircle of radius and one leg of length 3. Find the area of the triangle .
The answer can be expressed in the form of , where and are coprime positive integers . Enter .
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Let the other leg have length x . Then the tangents from Y and Z to the incircle have length ( x − 8 3 ) and ( 3 − 8 3 ) . So the hypotenuse has length x + 4 9 . The semiperimeter of the triangle is x + 8 2 1 , and the area of the triangle is 8 3 ( x + 8 2 1 ) . But the area can also be calculated as 2 3 x . Setting these expressions equal, we find x = 8 7 . Therefore,the area is equal to 1 6 2 1 . So, a = 2 1 and b = 1 6 . a + b = 3 7 .