Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.4 AMC 10/12 Find the answer

A circle of radius $r$ is inscribed in a right isosceles triangle, and a circle of radius $R$ is circumscribed about the triangle. Then $R/r$ equals

Pick one

Solution

Label the points as in the figure above. Let the side length $AB=AC=s$ . Therefore, $BC=s\sqrt{2}$ . Since the circumradius of a right triangle is equal to half of the length of the hypotenuse, we have $R=\frac{s\sqrt{2}}{2}$ .
Now to find the inradius. Notice that $IFAE$ is a square with side length $r$ , and also $AD=R$ . Therefore, $s=AD=AI+ID=r\sqrt{2}+r$ , and so $r=\frac{R}{\sqrt{2}+1}$ .
Finally, $\frac{R}{r}=1+\sqrt{2}, \boxed{\text{A}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.