Question

一个面积为 $4$ 的正方形内接于一个面积为 $5$ 的正方形，较小正方形的每个顶点分别在较大正方形的一条边上。较小正方形的一个顶点将较大正方形的一条边分成两段，长度分别为 $a$ 和 $b$ 。 $ab$ 的值是多少？

A square with area $4$ is inscribed in a square with area $5$ , with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$ and the other of length $b$ . What is the value of $ab ?$

$\dfrac{1}5$

$\dfrac{2}5$

$\dfrac{1}{2}$

$1$

$4$

答案：C

知识点：正方形（几何）全等（几何）面积分割

难度评级：1790

视频讲解：

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文字解答：

四个三角形都可以通过旋转互相得到，所以它们全等。因此可以如图标出 $a$ 。两个正方形之间的总面积为 $5-4 = 1$ ，所以有 $4$ 个全等三角形的总面积为 $1$ 。每个三角形面积为 $\dfrac{1}{4}$ 。每个三角形的面积也等于 $\frac {ab}2$ ，所以 $\frac{ab}2 = \dfrac{1}{4}$ 。因此 $ab = \dfrac{1}{2}$ 。

所以正确答案是 C。

Since all the triangles can be made from each other by rotating them around, they are all congruent. Therefore, we can place the $a$ as we have. The total area of the triangles is $5-4 = 1,$ so we have $4$ congruent triangles with a combined area of $1.$ This means the area of each triangle is $\dfrac{1}{4}.$ The area of each triangle is also $\frac {ab}2,$ so $\frac{ab}2 = \dfrac{1}{4}.$ This means $ab = \dfrac{1}{2} .$

Thus, the correct answer is C .

2012 AMC 8 第 25 题

答案：C

难度评级：1790

视频讲解：

文字解答：

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