2021 AMC 12B Problems/Problem 14

Problem

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$ . Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MACD?$

$\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}$

Solution

Solution 1

This question is just about pythagorean theorem $a^2+(a+2)^2-b^2 = (a+4)^2$ $2a^2+4a+4-b^2 = a^2+8a+16$ $a^2-4a+4-b^2 = 16$ $(a-2+b)(a-2-b) = 16$ $a=3, b=7$ With these calculation, we find out answer to be

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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All AMC 12 Problems and Solutions

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2021 AMC 12B Problems/Problem 14

Contents

Problem

Solution

Solution 1

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