2021 JMPSC Sprint Problems/Problem 18

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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 $a.$


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 $\boxed{21}$.


See also

  1. Other 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Sprint Answer Key
  3. All JMPSC Problems and Solutions

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