HMMT February

Geometry Difficulty 4.7 AIME Find the answer

Let $ABC$ be a right triangle with hypotenuse $AC$ . Let $B^{\prime}$ be the reflection of point $B$ across $AC$ , and let $C^{\prime}$ be the reflection of $C$ across $AB^{\prime}$ . Find the ratio of $[BCB^{\prime}]$ to $[BC^{\prime}B^{\prime}]$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $C, B^{\prime}$ , and $C^{\prime}$ are collinear, it is evident that $[BCB^{\prime}]=\frac{1}{2}[BCC^{\prime}]$ . It immediately follows that $[BCB^{\prime}]=[BC^{\prime}B^{\prime}]$ . Thus, the ratio is 1.

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