AMC8 2000

AMC8 2000 · Q18

AMC8 2000 · Q18. It mainly tests Area & perimeter, Coordinate geometry.

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

考虑这两个地理板上的四边形。以下哪个陈述是正确的？

(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。

(D) 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. 两个四边形面积相同，但I的周长小于II的周长。

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Divide each quadrilateral as shown. The resulting triangles each have base 1, altitude 1, and area $\frac{1}{2}$, so the quadrilaterals each have area 1. Three sides of quadrilateral I match those of quadrilateral II as indicated by matching marks. The fourth side of quadrilateral I is less than the fourth side of quadrilateral II, hence its perimeter is less, and choice (E) is correct.

答案（E）：如图将每个四边形分割。分割后得到的每个三角形底为 1，高为 1，面积为 $\frac{1}{2}$，所以每个四边形的面积都是 1。四边形 I 的三条边与四边形 II 的对应三条边相等（由相同记号表示）。四边形 I 的第四条边小于四边形 II 的第四条边，因此它的周长更小，所以选项（E）正确。

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