2018 AMC 12A 第 17 题

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

小提示：

把直角放在原点，使斜边为直线

3x + 4y = 12

。

Put the right angle at the origin so the hypotenuse is the line

3x + 4y = 12

大提示：

正方形远角

(s, s)

到该直线的距离为

2

；解

\tfrac{|3s + 4s - 12|}{5} = 2

并保留有效根

The far corner

(s, s)

of the square is distance

2

from that line; solve

\tfrac{|3s + 4s - 12|}{5} = 2

and keep the valid root

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

\frac{|3s + 4s - 12|}{\sqrt{3^2 + 4^2}} = \frac{|7s - 12|}{5} = 2\text{。}

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。

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{\frac{4}{49}}{6} = 1 - \frac{2}{147} = \frac{145}{147}\text{。}

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{\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

小提示：

大提示：

解答：

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