2018 AMC 10A 第 23 题

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}

设未种植正方形的边长为

x

。

Let

x

be the side length of the unplanted square

把大三角形分成正方形、两个直角三角形和一个高为

2

的三角形。

Decompose the triangle into the square, two right triangles, and a triangle of height

2

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}\text{。}

化简得

6 = \dfrac{7x}{2} + 5

x = \dfrac{2}{7}\text{。}

所求比例为

\dfrac{6 - x^2}{6} = \dfrac{6 - \frac{4}{49}}{6} = \dfrac{145}{147}\text{。}

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.

其他年份的第 23 题