Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular.  If $DP= 18$ and $EQ = 24$, then what is ${DF}$?
Medians intersect at the centroid, which is $1/3$ of the way along the median. Medians intersect at the centroid, which divides the medians in a 2:1 ratio. That means that PX is $\frac{1}{3}$ of DP and QX is $\frac{1}{3}$ of EQ. We can now use the Pythagorean Theorem to find DF. $DQ^2= DX^2 +QX^2$. $DQ^2= 12^2 + 8^2$. $DQ^2= 144 +64$. $DQ^2 = 208$. So $DQ = \sqrt{208}$. Finally, $DF=2DQ=2\sqrt{208}$. We can simplify that to $DF=4\sqrt{13}$. Right. So $DF=8\sqrt{13}$.

