AMC10 2009 B

AMC10 2009 B · Q24

AMC10 2009 B · Q24. It mainly tests Triangles (properties), Polygons.

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 $x$?

拱顶是古老的建筑特征。它由沿非平行边拼合的全等等腰梯形组成，如图所示。两端梯形的底边是水平的。用 9 个梯形构成的拱顶中，设 $x$ 为梯形较大内角的度数。求 $x$。

(A) 100 100

(B) 102 102

(D) 106 106

(E) 108 108

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Add a symmetric arch to the given arch to create a closed loop of trapezoids. Consider the regular 18-sided polygon created by the interior of the completed loop. Each interior angle of a regular 18-gon measures $$(18-2)\cdot 180^\circ/18 = 160^\circ.$$ Then $x + x + 160^\circ = 360^\circ$, so $x = 100^\circ$.

答案（A）：在给定的拱形旁添加一个对称的拱形，形成由梯形组成的闭合环。考虑由该闭合环内部形成的正十八边形。正十八边形的每个内角为 $$(18-2)\cdot 180^\circ/18 = 160^\circ.$$ 因此 $x + x + 160^\circ = 360^\circ$，所以 $x = 100^\circ$。

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