2021 JMPSC Sprint Problems/Problem 18

Problem

On square $ABCD$ with side length $28$ , $M$ is the midpoint of $\overline{CD}$ . Let $E$ be the foot of the altitude from $M$ to $\overline{AC}$ . If $AE$ can be represented as $a\sqrt{2}$ for some integer $a,$ find the value of $a.$

Solution

Notice that since $\angle ACD=45^\circ$ and $\angle MEC=90^\circ$ , $\triangle MEC$ is a 45-45-90 triangle. Thus, $EC=\frac{MC}{\sqrt{2}}=\frac{14}{\sqrt{2}}=7\sqrt{2}.$ Also, we have $AC=AD\sqrt{2}=28\sqrt{2}$ , so $AE=AC-EC=21\sqrt{2}$ which gives the answer of $\boxed{21}$ . ~tigerzhang

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2021 JMPSC Sprint Problems/Problem 18

Problem

Solution