Problem 89

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Problem 89: (2021 AMC 10a, 21)

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$? $\textbf{(A)} ~47\qquad$ $\textbf{(B)} ~52\qquad$ $\textbf{(C)} ~55\qquad$ $\textbf{(D)} ~58\qquad$ $\textbf{(E)} ~63$

Solution

Exercise 89: Solution

Let $ABCDEF$ be an equiangular hexagon. The lines $AB,CD,EF$ determine a triangle with area $48\sqrt3$, and the lines $BC,DE,FA$ determine a triangle with area $25\sqrt3$. The perimeter of hexagon $ABCDEF$ can be expressed as $m+n\sqrt p$, where $m,n,p$ are positive integers and $p$ is squarefree. What is $m+n+p$?

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