2019 AMC 8 第 21 题

Question

直线

y = 5

、

y = 1 + x

、

y = 1 - x

围成的三角形面积是多少？

What is the area of the triangle formed by the lines

y = 5,

y = 1 + x,

and

y = 1 - x?

4

8

10

12

16

答案：E

知识点：坐标几何三角形面积方程组

难度评级：1240

小提示：

找出三条直线两两相交的三个点

Find the three pairwise intersections

大提示：

用

y = 5

上的水平边作为底

Use the horizontal side on

y = 5

as the base

视频讲解：

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文字解答：

先求三角形的顶点。直线

y = 5

与另外两条直线的交点为

(4, 5)

和

(-4, 5)

。

求第三个交点时，令两条斜线相等，得到

1 + x = 1 - x \Rightarrow x = 0\text{。}

所以第三个顶点是

(0, 1)

。

前两个交点关于

y

轴对称，因此这个三角形是等腰三角形。

底边长为

2 \cdot 4 = 8

，高为

5 - 1 = 4

。三角形面积为

\dfrac{1}{2} \cdot 8 \cdot 4 = 16\text{。}

正确答案是 E。

First, let us find the vertices of the triangle. The intersections between

y = 5

and the other two lines are

(4, 5)

and

(-4, 5)

.

To find the third intersection, we can equate the two equations to get

1 + x = 1 - x \Rightarrow x = 0.

This gives us the point

(0, 1).

Note that the first two intersection points are mirrored across the

y

-axis. This tells us that the triangle is isosceles.

The base is therefore

2 \cdot 4 = 8,

and the height is

5 - 1 = 4.

The area of the triangle is therefore

\dfrac{1}{2} \cdot 8 \cdot 4 = 16.

Thus, the correct answer is E.