AMC12 2005 A

AMC12 2005 A · Q7

AMC12 2005 A · Q7. It mainly tests Area & perimeter, Coordinate geometry.

Square $EFGH$ is inside square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt{50}$, $E$ is between $B$ and $H$, and $BE = 1$. What is the area of the inner square $EFGH$?

正方形 $EFGH$ 在正方形 $ABCD$ 内部，使得 $EFGH$ 的每条边延长后都能通过 $ABCD$ 的一个顶点。正方形 $ABCD$ 的边长为 $\sqrt{50}$，$E$ 在 $B$ 和 $H$ 之间，且 $BE = 1$。内正方形 $EFGH$ 的面积是多少？

(A) 25 25

(B) 32 32

(D) 40 40

(E) 42 42

Answer

Correct choice: (C)

正确答案：（C）

Solution

Error creating thumbnail: Unable to save thumbnail to destination Arguable the hardest part of this question is to visualize the diagram. Since each side of $EFGH$ can be extended to pass through a vertex of $ABCD$, we realize that $EFGH$ must be tilted in such a fashion. Let a side of $EFGH$ be $x$. Error creating thumbnail: Unable to save thumbnail to destination Notice the right triangle (in blue) with legs $1, x+1$ and hypotenuse $\sqrt{50}$. By the Pythagorean Theorem, we have $1^2 + (x+1)^2 = (\sqrt{50})^2 \Longrightarrow (x+1)^2 = 49 \Longrightarrow x = 6$. Thus, $[EFGH] = x^2 = 36\ \mathrm{(C)}$

Error creating thumbnail: Unable to save thumbnail to destination 可以说这题最难的部分是想象图形。由于 $EFGH$ 的每条边都可以延长并通过 $ABCD$ 的一个顶点，我们知道 $EFGH$ 必须以这种方式倾斜放置。设 $EFGH$ 的边长为 $x$。 Error creating thumbnail: Unable to save thumbnail to destination 注意蓝色所示的直角三角形，其两条直角边分别为 $1$ 和 $x+1$，斜边为 $\sqrt{50}$。由勾股定理，$1^2 + (x+1)^2 = (\sqrt{50})^2 \Longrightarrow (x+1)^2 = 49 \Longrightarrow x = 6$。因此，$[EFGH] = x^2 = 36\ \mathrm{(C)}$

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