Question

农夫 Pythagoras 有一块直角三角形田地，两条直角边分别长 $3$ 和 $4$ 个单位。他在两条直角边相交的角落留出一个未种植的小正方形 $S$ ，使它从空中看起来像直角符号。田地的其余部分都已种植。 $S$ 到斜边的最短距离为 $2$ 个单位。田地中已种植部分占几分之几？

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted?

$\dfrac{25}{27}$

$\dfrac{26}{27}$

$\dfrac{73}{75}$

$\dfrac{145}{147}$

$\dfrac{74}{75}$

答案：D

知识点：面积分割直角三角形

难度评级：2150

解答：

设 $S$ 的边长为 $x$ ，可将田地分成下图所示的几个部分。

田地的面积可用两种方式表示： $\dfrac{3 \cdot 4}{2} = x^2 + \dfrac{x(3 - x)}{2}$ $+ \dfrac{x(4 - x)}{2} + \dfrac{2 \cdot 5}{2}.$

化简得 $6 = \dfrac{7x}{2} + 5$ $x = \dfrac{2}{7}.$

所求比例为 $\dfrac{6 - x^2}{6} = \dfrac{6 - \frac{4}{49}}{6} = \dfrac{145}{147}.$

因此正确答案是 D。

Let $x$ be the side length of $S.$ Then we can split the field up into the following shapes.

We can express the area of the field in two ways: $\dfrac{3 \cdot 4}{2} = x^2 + \dfrac{x(3 - x)}{2}$ $+ \dfrac{x(4 - x)}{2} + \dfrac{2 \cdot 5}{2}.$

Simplifying yields $6 = \dfrac{7x}{2} + 5$ $x = \dfrac{2}{7}.$

The desired fraction is $\dfrac{6 - x^2}{6} = \dfrac{6 - \frac{4}{49}}{6} = \dfrac{145}{147}.$

Thus, D is the correct answer.

2018 AMC 10A 第 23 题

答案：D

难度评级：2150

解答：

其他年份的第 23 题