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

See also

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

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. JMPSC.png