Answer
The lengths of the medians are $\sqrt{37}$, $\displaystyle \frac{\sqrt{109}}{2},\ \displaystyle \frac{\sqrt{145}}{2}$.
Work Step by Step
The midpoint of $AB$ is $M=(\displaystyle \frac{1+3}{2}, \displaystyle \frac{0+6}{2})=(2,3)$.
The length of the median $CM$ is
$d(C, M)=\sqrt{(2-8)^{2}+(3-2)^{2}}=\sqrt{37}$.
The midpoint of $AC$ is $N=(\displaystyle \frac{1+8}{2}, \displaystyle \frac{0+2}{2})=(\frac{9}{2},1)$.
The length of the median $BN$ is
$d(B, N)=\displaystyle \sqrt{(\frac{9}{2}-3)^{2}+(1-6)^{2}}=\frac{\sqrt{109}}{2}$.
The midpoint of $BC$ is $K=(\displaystyle \frac{3+8}{2}, \displaystyle \frac{6+2}{2})=(\frac{11}{2},4)$.
The length of the median $AK$ is
$d(A, K)=\displaystyle \sqrt{(\frac{11}{2}-1)^{2}+(4-0)^{2}}=\frac{\sqrt{145}}{2}$.
