Too small radius!

Geometry Level pending

Right triangle X Y Z , XYZ, with hypotenuse Y Z , YZ, has an incircle of radius 3 8 \dfrac{3}{8} and one leg of length 3. Find the area of the triangle .

The answer can be expressed in the form of a b \dfrac{a}{b} , where a a and b b are coprime positive integers . Enter a + b a+b .


The answer is 37.

Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Ayush G Rai
Jul 28, 2016

Let the other leg have length x . x. Then the tangents from Y Y and Z Z to the incircle have length ( x − 3 8 ) (x-\frac{3}{8}) and ( 3 − 3 8 ) . (3-\frac{3}{8}). So the hypotenuse has length x + 9 4 . x+\frac{9}{4}. The semiperimeter of the triangle is x + 21 8 , x+\frac{21}{8}, and the area of the triangle is 3 8 ( x + 21 8 ) . \frac{3}{8}(x+\frac{21}{8}). But the area can also be calculated as 3 x 2 . \frac{3x}{2}. Setting these expressions equal, we find x = 7 8 . x =\frac{7}{8}. Therefore,the area is equal to 21 16 . \frac{21}{16}. So, a = 21 a=21 and b = 16. a + b = 37 . b=16.a+b=\boxed{37}.

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