2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 5 Easy

Square ABCD has side length 1. Point P, Q, R, and S each lie on a side of ABCD such that APQCRS is an equilateral convex hexagon with side length s. What is s?

  • A.

    \frac{\sqrt{2}}{3}

  • B.

    \frac{1}{2}

  • C.

    2-\sqrt{2}

  • D.

    1-\frac{\sqrt{2}}{4}

  • E.

    \frac{2}{3}

Answer:C

Assume BP=x.

AP=PQ, PB=BQ,

\triangle {BPQ} is an isosceles right triangle.

\sqrt{2} x=1-x

(1+\sqrt{2})x=1

x=\frac{1}{1+\sqrt{2}}

AP=\frac{\sqrt{2}}{1+\sqrt{2}}=2-\sqrt{2}.

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