2023 AMC 10A Problems/Problem 17

Problem

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$ . Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$ ?

$\textbf{(A) } 84 \qquad \textbf{(B) } 86 \qquad \textbf{(C) } 88 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 92$

Solution

[insert asy diagram]

Using knowledge of common Pythagorean triples and guess and check, we can find that $\triangle{ABP}$ is a $8$ - $15$ - $17$ triangle with side lengths $16$ - $30$ - $34$ , and $\triangle{PCQ}$ and $\triangle{QDA}$ are $3$ - $4$ - $5$ triangles with side lengths $9$ - $12$ - $15$ and $21$ - $28$ - $35$ , respectively.

Adding up the side lengths of $\triangle{APQ}$ gives $34+15+35=\boxed{\textbf{(A) } 84}.$

~ItsMeNoobieboy

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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All AMC 10 Problems and Solutions

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2023 AMC 10A Problems/Problem 17

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