Question

Farmer Pythagoras 有一块直角三角形田地。这个直角三角形的两条直角边长分别为 $3$ 和 $4$ 个单位。在两边相交的直角角落里，他留下一小块未种植的正方形 $S$ ，使得从空中看像直角标记。田地其余部分都已种植。 $S$ 到斜边的最短距离是 $2$ 个单位。田地中已种植的部分占多少比例？

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths of $3$ 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?

$\tfrac{25}{27}$

$\tfrac{26}{27}$

$\tfrac{73}{75}$

$\tfrac{145}{147}$

$\tfrac{74}{75}$

答案：D

知识点：坐标几何距离公式面积

难度评级：1910

解答：

将直角放在原点，两条直角边沿坐标轴，则顶点为 $(4, 0)$ 、 $(0, 3)$ 、 $(0, 0)$ ，正方形 $S$ 为 $[0, s] \times [0, s]$ 。斜边是 $3x + 4y - 12 = 0$ ，它到正方形最近的角 $(s, s)$ 的距离满足这给出 $s = \tfrac{22}{7}$ 或 $s = \tfrac27$ ；只有 $s = \tfrac27$ 能使正方形留在三角形内部。 $\frac{|3s + 4s - 12|}{\sqrt{3^2 + 4^2}} = \frac{|7s - 12|}{5} = 2.$

田地面积为 $\tfrac12 \cdot 3 \cdot 4 = 6$ ，未种植正方形面积为 $\left(\tfrac27\right)^2 = \tfrac{4}{49}$ 。已种植比例为 $1 - \frac{4/49}{6} = 1 - \frac{2}{147} = \frac{145}{147}.$

所以正确答案是 D。

Place the right angle at the origin with legs on the axes, so the vertices are $(4, 0),$ $(0, 3),$ $(0, 0),$ and the square $S$ is $[0, s] \times [0, s].$ The hypotenuse is $3x + 4y - 12 = 0,$ and the distance from its nearest corner $(s, s)$ is $\frac{|3s + 4s - 12|}{\sqrt{3^2 + 4^2}} = \frac{|7s - 12|}{5} = 2.$ This gives $s = \tfrac{22}{7}$ or $s = \tfrac27;$ only $s = \tfrac27$ keeps the square inside the triangle.

The field has area $\tfrac12 \cdot 3 \cdot 4 = 6$ and the unplanted square has area $\left(\tfrac27\right)^2 = \tfrac{4}{49}.$ The planted fraction is $1 - \frac{4/49}{6} = 1 - \frac{2}{147} = \frac{145}{147}.$

Thus, the correct answer is D.

2018 AMC 12A 第 17 题

答案：D

难度评级：1910

解答：

其他年份的第 17 题