2023 AMC 12B Problems/Problem 3

Problem

A 3-4-5 right triangle is inscribed circle $A$ , and a 5-12-13 right triangle is inscribed in circle $B$ . What is the ratio of the area of circle $A$ to circle $B$ ?

$\textbf{(A) }\frac{9}{25}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{5}\qquad\textbf{(D) }\frac{25}{169}\qquad\textbf{(E) }\frac{4}{25}$

Solution 1

Because the triangle are right triangles, we know the hypotenuses are diameters of circles $A$ and $B$ . Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply $\pi$ to get $6.25\pi$ and $42.25\pi$ as the areas of the circles. Multiply 4 on both numbers to get $25\pi$ and $169\pi$ . Cancel out the $\pi$ , and lastly, divide, to get your answer $\boxed{\textbf(D)\frac{25}{169}}$ .

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2023 AMC 12B Problems/Problem 3

Problem

Solution 1