2024 AMC 12A Problems/Problem 10

Problem

Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$ , what is $\beta$ ?

$\textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad \textbf{(D) }\frac{\alpha}{2}\qquad \textbf{(E) }\pi - 4\alpha\qquad$

Solution 1

From question, $tan\alpha=\frac{3}{4}, \space tan\beta=\frac{7}{24}$ $tan(\alpha+\beta)= \frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}$ $tan(\alpha+\beta)= \frac{\frac{3}{4}+\frac{7}{24}}{1-\frac{3}{4} \cdot \frac{7}{24}}$ $tan(\alpha+\beta)=\frac{4}{3}$ $\alpha+\beta=tan^{-1}(\frac{4}{3})$ $\alpha+\beta=\frac{\pi}{2}-\alpha$ $$ (Error compiling LaTeX. Unknown error_msg)\beta=\fbox{(C) $\frac{\pi}{2} -2\alpha$ }$$ (Error compiling LaTeX. Unknown error_msg)

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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All AMC 12 Problems and Solutions

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2024 AMC 12A Problems/Problem 10

Problem

Solution 1

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