### Instruction: Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular.  If $DP= 18$ and $EQ = 24$, then what is ${DE}$?### Response: I recall that the centroid of a triangle, where the three medians intersect, divides each median into a ratio of 2:1, with the longer segment closer to the vertex.
So, if I let $G$ be the centroid of $\triangle DEF$, then I have $DG = 12$, $GP = 6$, $EG = 16$, and $GQ = 8$.
Since $\overline{DP}$ and $\overline{EQ}$ are perpendicular, $\triangle DGE$ is a right triangle, and I can use the Pythagorean theorem to find $DE$.
I have $DE^2 = DG^2 + EG^2$, so $DE^2 = 12^2 + 16^2 = 144 + 256 = 400$.
Taking the square root of both sides, I get $DE = 20$.