AMC8 2002

AMC8 2002 · Q15

AMC8 2002 · Q15. It mainly tests Area & perimeter, Counting in geometry (lattice points).

Which of the following polygons has the largest area?

下列哪个多边形面积最大？

(A) A A

(B) B B

(D) D D

(E) E E

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) Areas may be found by dividing each polygon into triangles and squares as shown. Note: Pick’s Theorem may be used to find areas of geoboard polygons. If $I$ is the number of dots inside the figure, $B$ is the number of dots on the boundary and $A$ is the area, then $A = I + \frac{B}{2} - 1$. Geoboard figures in this problem have no interior points, so the formula simplifies to $A = \frac{B}{2} - 1$. For example, in polygon $D$ the number of boundary points is 11 and $\frac{11}{2} - 1 = 4\frac{1}{2}$.

（E）如图所示，可以通过把每个多边形分割成三角形和正方形来求面积。注：可以用皮克定理来求钉板（geoboard）多边形的面积。若 $I$ 为图形内部点的个数，$B$ 为边界上的点的个数，$A$ 为面积，则 $A = I + \frac{B}{2} - 1$。本题中的钉板图形没有内部点，因此公式可简化为 $A = \frac{B}{2} - 1$。例如，在多边形 $D$ 中，边界点的个数为 11，且 $\frac{11}{2} - 1 = 4\frac{1}{2}$。

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