AMC 8 2000 Problem 18
AMC8 2000 年第 18 题
面积与周长（综合）★★★☆☆
Consider these two geoboard quadrilaterals. Which of the following statements is true?
考虑下面几何画板上的两个四边形。下面哪句话是对的？
- A.The area of quadrilateral I is more than the area of quadrilateral II. / 四边形 I 的面积大于四边形 II 的面积。
- B.The area of quadrilateral I is less than the area of quadrilateral II. / 四边形 I 的面积小于四边形 II 的面积。
- C.The quadrilaterals have the same area and the same perimeter. / 两个四边形的面积相同，周长也相同。
- D.The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II. / 两个四边形的面积相同，但 I 的周长大于 II 的周长。
- E.The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II. / 两个四边形的面积相同，但 I 的周长小于 II 的周长。
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
First consider the area of the two figures. Assume that the pegs are 1 unit apart. Divide region I into two triangles by drawing a horizontal line on the second row from the top. Shifting the bottom triangle up one unit will create a square, and this square has the same area as region I. Thus, region I has an area of 1 square unit. Draw a horizontal line on figure II on the fourth row from the top to divide the figure into two triangles. Temporarily move the top of the higher triangle one peg to the left. This will preserve the area of the triangle, as you are keeping both the height and the base of the triangle equal. Now you have two congruent triangles that are half of the area of a unit square. Thus, region II also has an area of 1 square unit, and the areas are equal. To compare the perimeters, note that region I has two different side lengths: two sides are 1 unit apart, and the other two sides are apart, as they are the diagonal of a unit square. The total perimeter is .
Using the Pick's Theorem, The area of both figures is 1. This rules out answer choices A and B. When using the Pythagorean Theorem, we find that the perimeter of I is 2 + 2( ) and the perimeter of II is 1 + 2( ) + . Since > 1, the perimeter of II is greater, therefore the correct answer is . Yaay!
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