A geometry problem by ritwik jain

Geometry Level 5

Let $AD$ be an altitude in a right triangle $ABC$ with $\angle A = 90^\circ$ and $D$ on $BC$ . Suppose that the radii of the incircles of the $\triangle ABD$ and $\triangle ACD$ are 33 and 56 respectively. Let $r$ be the radius of the incircle of the $\triangle ABC$ . Find the value of $r$ .

The answer is 65.

1 solution

Maria Kozlowska
Oct 14, 2017

$\triangle ABD$ and $\triangle ACD$ are similar.

Let $AB=c, AC=b$ .

$\dfrac{b}{c}=\dfrac{56}{33}$

$\dfrac{\sqrt{b^2+c^2}}{c}=\dfrac{r}{33}$

$\sqrt{b^2/c^2+1}=\dfrac{r}{33}$

$\sqrt{56^2/33^2+1}=\dfrac{r}{33}$

$r=\boxed{65}$

A geometry problem by ritwik jain

The answer is 65.

1 solution

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