Question

一个矩形的对角线长为 $x$ ，长是宽的两倍。该矩形面积是多少？

A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?

$\dfrac{1}{4}x^2$

$\dfrac{2}{5}x^2$

$\dfrac{1}{2}x^2$

$x^2$

$\dfrac{3}{2}x^2$

设宽为 $w$ ，则长为 $2w$ 。由勾股定理， $x^2 = w^2 + (2w)^2 = 5w^2.$

由对角线关系可知，面积为 $w \cdot 2w = 2w^2 = \dfrac{2}{5}x^2$ 。

所以正确答案是 B。

Let the width be $w.$ Then the length is $2w,$ and the diagonal gives $x^2 = w^2 + (2w)^2 = 5w^2.$

The area is $w \cdot 2w = 2w^2 = \dfrac{2}{5}x^2.$

Thus, the correct answer is B.