Problem 152

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Problem 152: (2021 AMC 12b, 24)

Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ $\textbf{(A) }81 \qquad$ $ \textbf{(B) }89 \qquad$ $ \textbf{(C) }97\qquad$ $ \textbf{(D) }105 \qquad$ $ \textbf{(E) }113$

Solution

Exercise 152: Solution

Let $ABCD$ be a parallelogram with area $6$. Points $P,Q$ are the projections of $A,C$ onto line $BD$, and $R,S$ are the projections of $B,D$ onto line $AC$. Suppose $PQ=4$ and $RS=6$, and let $d$ denote the length of $\overline{BD}$, the longer diagonal. Then $d^2$ can be written as $m+n\sqrt p$. What is $m+n+p$?

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