1976 AMC 12 第 14 题

Question

一个凸多边形的各内角度数成等差数列。若最小角为

100^\circ

，最大角为

140^\circ

，则该多边形的边数为

The measures of the interior angles of a convex polygon are in arithmetic progression. If the smallest angle is

100^\circ

and the largest angle is

140^\circ,

then the number of sides the polygon has is

6

8

10

11

12

等差数列的平均值等于首项与末项的平均值

The average of an arithmetic progression is the average of its first and last terms

令

120n

等于内角和

180(n-2)

Equate

120n

with the interior-angle sum

180(n-2)

平均角度为

\frac{100+140}{2}=120^\circ

，所以内角和为

120n

。凸

n

边形的内角和为

180(n-2)

，故

120n=180(n-2)

，解得

n=6

。

所以正确答案是 A。

The average angle is

\frac{100+140}{2}=120^\circ,

so the angle sum is

120n.

A convex

n

-gon has angle sum

180(n-2),

hence

120n=180(n-2)

and

n=6.

Therefore, the correct answer is A.

其他年份的第 14 题