Question

楔石拱是一种古老的建筑结构。它由全等的等腰梯形沿非平行边拼合而成，如图所示。两端梯形的底边是水平的。一个由 $9$ 个梯形组成的拱中，设 $x$ 为梯形较大内角的度数。求 $x$ 的值。

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x?$

$100$

$102$

$104$

$106$

$108$

答案：A

知识点：正多边形角度和对称性

难度评级：1600

解答：

加上镜像后，拱补成由 $18$ 个梯形组成的对称闭环。它们的内侧边形成正 $18$ 边形，每个内角为 $\dfrac{(18-2)\cdot180^\circ}{18}=160^\circ.$

在每个内侧顶点，两个梯形的较大内角 $x$ 与 $160^\circ$ 之和为一周角，因此 $x+x+160^\circ=360^\circ$ ，所以 $x=100$ 。

所以正确答案是 A。

Adding a mirror image completes the arch into a symmetric closed loop of $18$ trapezoids. Their inner edges form a regular $18$ -gon, each interior angle of which is $\dfrac{(18-2)\cdot180^\circ}{18}=160^\circ.$

At each inner vertex, two of the trapezoids' larger angles $x$ meet the $160^\circ$ angle around a full turn: $x+x+160^\circ=360^\circ,$ so $x=100.$

Thus, the correct answer is A.

2009 AMC 10B 第 24 题

答案：A

难度评级：1600

解答：

其他年份的第 24 题