Question

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

$\dfrac{\alpha}{3}$

$\alpha-\dfrac{\pi}{8}$

$\dfrac{\pi}{2}-2\alpha$

$\dfrac{\alpha}{2}$

$\pi-4\alpha$

Answer: C

Concepts:trigonometric identity right triangle

Difficulty rating: 1570

Solution:

The smallest angle of the $3\text{-}4\text{-}5$ triangle has $\tan\alpha=\tfrac34.$ Then $\begin{aligned} &\tan2\alpha=\frac{2\cdot\frac34}{1-\frac{9}{16}} \\ &=\frac{3/2}{7/16}=\frac{24}{7}. \end{aligned}$ The smallest angle of the $7\text{-}24\text{-}25$ triangle has $\tan\beta=\tfrac{7}{24}=\cot2\alpha$ $=\tan\!\left(\tfrac{\pi}{2}-2\alpha\right).$ Hence $\beta=\tfrac{\pi}{2}-2\alpha.$ Thus, the correct answer is C.

2024 AMC 12A Problem 10

Answer: C

Difficulty rating: 1570

Solution:

Problem 10 in Other Years