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

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 $2 \cdot \dfrac{1}{2} \cdot 1 \cdot 1 = 1.$ 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 $1,$ makings 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

解答：

其他年份的第 18 题