Math Gold Medalist

Lor

2023 AMC 10A

Problem 17

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 $\triangle{APQ}$?

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

Important Identities

Pythagorean Theorem

Parity

Prime Factorization

Casework

Apply Pythagorean Theorem twice and then use a^2-b^2=(a-b)(a+b)

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   Solution