AMC12 2017 A

AMC12 2017 A · Q11

AMC12 2017 A · Q11. It mainly tests Triangles (properties), Polygons.

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 2017. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

Claire 将一个凸多边形的内角度数相加，得到和为 2017。然后她发现忘记包含了一个角。被遗忘的角的度数是多少？

(A) 37 37

(B) 63 63

(D) 143 143

(E) 163 163

Answer

Correct choice: (D)

正确答案：（D）

Solution

If the polygon has $n$ sides and the degree measure of the forgotten angle is $\alpha$, then $(n-2)180=2017+\alpha$. Because $0<\alpha<180$, $2017<(n-2)180<2197$, which implies that $n=14$, the angle sum is 2160, and $\alpha=143$. To see that such a polygon exists, draw a circle and a central angle of measure $143^\circ$, and divide the minor arc spanned by the angle into 12 small arcs. The polygon is then formed by the two radii and 12 small chords, as illustrated.

如果多边形有 $n$ 条边，被遗忘角的度数为 $\alpha$，则 $(n-2)180=2017+\alpha$。因为 $0<\alpha<180$，$2017<(n-2)180<2197$，这意味着 $n=14$，角和为 2160，$\alpha=143$。为了看到这样的多边形存在，画一个圆和一个 $143^\circ$ 的中心角，将该角跨越的小弧分成 12 个小弧。然后由两条半径和 12 条小弦形成多边形，如图所示。

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