Difference between revisions of "2023 AMC 12B Problems/Problem 3"

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<math>\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}</math>
 
<math>\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}</math>
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==Solution 1==
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Because the triangle are right triangles, we know the hypotenuses are diameters of circles <math>A</math> and <math>B</math>. Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply <math>\pi</math> to get <math>6.25\pi</math> and <math>42.25\pi</math> as the areas of the circles. Multiply 4 on both numbers to get <math>25\pi</math> and <math>169\pi</math>. Lastly, divide, to get your answer: <math>\frac{25}{169}</math> = <math>\boxed{\textbf{(D) \frac{25}{169}}</math>.
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~zhenghua
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~Failure.net

Revision as of 13:25, 15 November 2023

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 $169\pi$. Lastly, divide, to get your answer: $\frac{25}{169}$ = $\boxed{\textbf{(D) \frac{25}{169}}$ (Error compiling LaTeX. Unknown error_msg).

~zhenghua ~Failure.net