Question

图中大正方形的每条边都被三等分。内接正方形的顶点在这些三等分点上，如图所示。内接正方形面积与大正方形面积的比是

Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is

$\dfrac{\sqrt{3}}{3}$

$\dfrac{5}{9}$

$\dfrac{2}{3}$

$\dfrac{\sqrt{5}}{3}$

$\dfrac{7}{9}$

答案：B

知识点：面积比勾股定理

难度评级：1330

解答：

设大正方形边长为 $3x$ 。由勾股定理，内接正方形边长为 $\sqrt{(2x)^2 + x^2} = \sqrt{5x^2} = x\sqrt{5}$

大正方形面积为内接正方形面积为 $(3x)^2 = 9x^2$ $(x\sqrt{5})^2 = 5x^2.$

因此面积比为 $\dfrac{5}{9}$ 。

所以正确答案是 B。

Let $3x$ be the side length of the large square. Then we can find the side length of the inner square via $\sqrt{(2x)^2 + x^2} = \sqrt{5x^2} = x\sqrt{5}$ from the Pythagorean Theorem.

The area of the larger square is $(3x)^2 = 9x^2$ and that of the inner square is $(x\sqrt{5})^2 = 5x^2.$

The ratio of the areas is then $\dfrac{5}{9}.$

Thus, B is the correct answer.

1997 AMC 8 第 15 题

答案：B

难度评级：1330

解答：

其他年份的第 15 题