# 2023 AMC 12B Problems/Problem 3

The following problem is from both the 2023 AMC 10B #3 and 2023 AMC 12B #3, so both problems redirect to this page.

## 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 $B$?

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

## Solution 1

Because the triangles 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 $=\boxed{\textbf{(D) }\frac{25}{169}}.$

~Failure.net

## Solution 1.5 (Way FASTER! Use in the contest)

With right triangles inscribed in circles, the hypotenuse must be the diameter. Therefore, the ratio of radii is $\frac{5}{13}$ which means the area ratio is just $\frac{5^2}{13^2}$ so, $\boxed{\textbf{(D) }\frac{25}{169}}.$

-Mismatchedcubing/Andrew_Lu

## Solution 2

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 $=\boxed{\textbf{(D) }\frac{25}{169}}.$

~Mintylemon66

## Solution 3

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

~vsinghminhas

~Math-X

## Video Solution (Quick and Easy!)

~Education, the Study of Everything

## Video Solution

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

## Video Solution by Interstigation

 2023 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 2 Followed byProblem 4 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions
 2023 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 2 Followed byProblem 4 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions