2024 AMC 12A Problem 10

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

\tan\alpha=\dfrac34;

apply the double-angle formula for

\tan2\alpha

\tan\beta=\dfrac{7}{24}

is the reciprocal of

\tan2\alpha

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{\frac{3}{2}}{\frac{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.

Problem 10 in other years