Problem 246

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Problem 246: (2021 AMC 12b, 11)

Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$

$\textbf{(A) }\frac{42}5 \qquad$ $ \textbf{(B) }6\sqrt2 \qquad$ $ \textbf{(C) }\frac{84}5\qquad$ $ \textbf{(D) }12\sqrt2 \qquad$ $ \textbf{(E) }18$

Solution

Exercise 246: Solution

Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=6$ (so $AP=9$). There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$

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