1985 AMC 12 第 13 题

Question

木板上的钉子在水平和竖直方向都相距

1

个单位。如图，将一条橡皮筋套在

4

个钉子上，形成一个四边形。它的面积为多少平方单位？

Pegs are put in a board

1

unit apart both horizontally and vertically. A rubber band is stretched over

4

pegs as shown in the figure, forming a quadrilateral. Its area in square units is

4

4.5

5

5.5

6

给四个顶点处的钉子赋予整数坐标

Assign integer coordinates to the four corner pegs

对

(1,3)

、

(4,1)

、

(3,0)

、

(0,1)

使用鞋带公式

Use the shoelace formula on

(1,3),

(4,1),

(3,0),

and

(0,1)

以左下角的钉子为

(0,0)

，四边形按顺序的顶点为

(1,3)

、

(4,1)

、

(3,0)

、

(0,1)

。鞋带公式给出

\begin{aligned} &\frac12\left|(1+0+3+0)\right.\\ &\quad\left.{}-(12+3+0+1)\right|=6\text{。} \end{aligned}

所以正确答案是 E。

With the lower-left peg as

(0,0),

the quadrilateral’s vertices in order are

(1,3),

(4,1),

(3,0),

and

(0,1).

The shoelace formula gives

\begin{aligned} &\frac12\left|(1+0+3+0)\right.\\ &\quad\left.{}-(12+3+0+1)\right|=6. \end{aligned}

Therefore the correct answer is E.

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