2000 AMC 8 第 18 题

Question

考虑这两个钉板四边形。下列哪一项正确？

Consider these two geoboard quadrilaterals. Which of the following statements is true?

四边形

I

的面积大于四边形

II

的面积。

The area of quadrilateral

I

is more than the area of quadrilateral

II.

四边形

I

的面积小于四边形

II

的面积。

The area of quadrilateral

I

is less than the area of quadrilateral

II.

两个四边形面积相同，周长也相同。

The quadrilaterals have the same area and the same perimeter.

两个四边形面积相同，但

I

的周长大于

II

的周长。

The quadrilaterals have the same area, but the perimeter of

I

is more than the perimeter of

II.

两个四边形面积相同，但

I

的周长小于

II

的周长。

The quadrilaterals have the same area, but the perimeter of

I

is less than the perimeter of

II.

答案：E

知识点：面积周长格点

难度评级：1470

小提示：

将两个四边形都分解成单位直角三角形。

Decompose both quadrilaterals into unit right triangles

大提示：

计算每个四边形四条边的长度。

Compute all four side lengths of each quadrilateral

解答：

假设这个网格上相邻钉点相距

1

个单位。

区域

I

是底为

1

、高为

1

的平行四边形，所以面积为

1 \cdot 1 = 1

。

区域

II

可以分成

2

个三角形，每个底为

1

、高为

1

。面积总和为

2 \cdot \dfrac{1}{2} \cdot 1 \cdot 1 = 1\text{。}

因此两个区域面积相同。

每个区域都有

2

条长度为

\sqrt{2}

的边。区域

I

有

2

条单位边，而区域

II

只有

1

条单位边。

区域

II

的另一条边显然大于

1

，所以区域

II

的周长更大。

所以正确答案是 E。

Assume that the pegs on this grid are separated by

1

unit.

Note that region

I

is a parallelogram with base

1

and height

1,

making its area

1 \cdot 1 = 1.

We can split region

II

into

2

triangles, both with base

1

and height

1.

This makes the sum of the areas

2 \cdot \dfrac{1}{2} \cdot 1 \cdot 1 = 1.

This shows that both regions have the same area.

Note that each region has

2

sides that are of length

\sqrt{2}.

Region

I

has

2

unit sides, whereas region

II

only has

1.

The other side of region

II

is clearly greater than

1,

which shows that region

II

has the greater perimeter.

Thus, E is the correct answer.

2000 AMC 8 第 18 题

答案：E

难度评级：1470

小提示：

大提示：

解答：

其他年份的第 18 题