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2009 AMC 10B Problem 24

Problem 24 of 25HarderGeometry

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 99 trapezoids, let xx be the angle measure in degrees of the larger interior angle of the trapezoid. What is x?x?

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Solution

Adding a mirror image completes the arch into a symmetric closed loop of 1818 trapezoids. Their inner edges form a regular 1818-gon, each interior angle of which is (18−2)⋅180∘18=160∘. \dfrac{(18-2)\cdot180^\circ}{18}=160^\circ. At each inner vertex, two of the trapezoids’ larger angles xx meet the 160∘160^\circ angle around a full turn: x+x+160∘=360∘,x+x+160^\circ=360^\circ, so x=100.x=100. Thus, the correct answer is A.

More practice

Concepts: regular polygon · angle sum · symmetry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.