FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3

Problem

There are $\binom{8}{4}=70$ distinct quadrilaterals that can be formed from the vertices of a regular octagon. Which of these statements must hold true for all those quadrilaterals?

$\textbf{(A) }\text{All of the 70 quadrilaterals are rectangles.}\\ \textbf{(B) }\text{All of the 70 quadrilaterals are cyclic.}\\ \textbf{(C) }\text{All of the 70 quadrilaterals are trapezoids.}\\ \textbf{(D) }\text{All of the 70 quadrilaterals are kites.}\\ \textbf{(E) }\text{None of the above statements are true.}$

Solution

All regular octagons can be inscribed in a circle, thus any subset of $4$ vertice's from this octagon also all lie on the same circle. It is easy to see that none of the other answer choices work. Therefore, the answer is $\boxed{\textbf{(B) }\text{All of the 70 quadrilaterals are cyclic.}}$

FidgetBoss 4000's 2019 Mock AMC 12B (Problems • Answer Key • Resources)
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FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3

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