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Problem 780

AIME late

Geometry Difficulty 5.5 Find the answer

The area of the large square $\mathrm{ABCD}$ is 18 square centimeters, and the side $\mathrm{MN}$ of the gray square $\mathrm{MNPQ}$ lies on the diagonal $\mathrm{BD}$ , with vertex $\mathrm{P}$ on side $\mathrm{BC}$ and $\mathrm{Q}$ on side $\mathrm{CD}$ . What is the area of the gray square $\mathrm{MNPQ}$ in square centimeters?

Your answer

Official solution

10. Answer: 4
Analysis: Connect $\mathrm{AC}$ intersecting BD at 0, and construct the circumscribed square $\mathrm{EFGH}$ of the larger square $\mathrm{ABCD}$ , as shown in the figure. Then, the area of square $\mathrm{EFGH}$ is 36 square centimeters. Therefore, $\mathrm{DB}=\mathrm{AC}=6$ centimeters.
It is easy to see that $D M=M Q=M N=N B=2$ centimeters.
Thus, the area of the shaded square is 4 square centimeters.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.