Question

Daniel 找到一张长方形索引卡片，量得其对角线为 $8$ 厘米。然后他在卡片的两个相对角各剪去一个边长为 $1$ 厘米的正方形，并量得这两个正方形最近的两个顶点之间的距离为 $4 \sqrt{2}$ 厘米，如下图所示。原索引卡片的面积是多少？

Daniel finds a rectangular index card and measures its diagonal to be $8$ centimeters. Daniel then cuts out equal squares of side $1$ cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4 \sqrt{2}$ centimeters, as shown below. What is the area of the original index card?

$14$

$10 \sqrt{2}$

$16$

$12 \sqrt{2}$

$18$

答案：E

知识点：矩形勾股定理代数变形

难度评级：1370

解答：

如图，将原长方形的宽和高分别记为 $a$ 和 $b$ 。于是有和 $a^2 + b^2 = 64$ $(a - 2)^2 + (b - 2)^2 = 32.$

后一式化简为也就是因此 $a^2 + b^2 - 4a - 4b + 4 + 4 = 32,$ $72 - 4(a + b) = 32.$ $a + b = 10.$

将它平方，得到从而因此面积 $(ab)$ 为 $18$ 。 $a^2 + b^2 + 2ab = 100,$ $2ab = 36,$

所以正确答案是 E。

We can label $a$ and $b$ as the width and height as in the diagram. Then we get that $a^2 + b^2 = 64$ and $(a - 2)^2 + (b - 2)^2 = 32.$

The latter expression simplifies to $a^2 + b^2 - 4a - 4b + 4 + 4 = 32,$ which is the same as $72 - 4(a + b) = 32.$ From this we get $a + b = 10.$

Squaring this, we get $a^2 + b^2 + 2ab = 100,$ which gets us that $2ab = 36,$ which means that the area $(ab)$ is $18.$

Thus, E is the correct answer.

2022 AMC 10A 第 10 题

答案：E

难度评级：1370

解答：

其他年份的第 10 题