2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 16 Easy

The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

  • A.

    15\dfrac{1}{8}

  • B.

    15\dfrac{3}{8}

  • C.

    15\dfrac{1}{2}

  • D.

    15\dfrac{5}{8}

  • E.

    15\dfrac{7}{8}

Answer:D

We first notice that the side length of the square is 5 by Pythagorean Theorem. Then, by Triple Perpendicular Model, the four right triangles are similar to each other. A_{\triangle A B C}: A_{\triangle C D E}: A_{\triangle E F G}: \text{top triangle}=1: 1: \frac{1}{16}: \frac {25}{16}. The area of the top right triangle is therefore \frac{25}{16}\times 6=\frac {75}{8}. Hence the shaded area is 25 - \frac{75}{8}=\frac{125}{8}=15\frac{5}{8}.

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