2023 AMC 10B Problems/Problem 3

Problem

A $3-4-5$ right triangle is inscribed in 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 the area of circle $B$ ?

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

Solution 1

Since the arc angle of the diameter of a circle is $90$ degrees, the hypotenuse of each these two triangles is respectively the diameter of circles $A$ and $B$ .

Therefore the ratio of the areas equals the radius of circle $A$ squared : the radius of circle $B$ squared $=$ $0.5\times$ the diameter of circle $A$ , squared : $0.5\times$ the diameter of circle $B$ , squared $=$ the diameter of circle $A$ , squared: the diameter of circle $B$ , squared $=\boxed{\textbf{(B) }\frac{25}{169}}.$

~Mintylemon66

Solution 2

The ratio of areas of circles is the same as the ratios of the diameters squared. Since this is a right triangle the hypotenuse of each triangle will be the diameter of the circle. This yields the expression $\frac{5^2}{13^2} =\boxed{\textbf{(B) }\frac{25}{169}}.$

~vsinghminhas

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2023 AMC 10B Problems/Problem 3

Problem

Solution 1

Solution 2