Question

In the given figure, ${AB}^2+{AC}^2=122$, $BC=10$. Find the length of the median on side $BC$.

Answer 1

In $\Delta ABC$, seg $AQ$ is a median
$\therefore$ $BQ=\frac{1}{2}BC=\frac{1}{2}\times 10$
$BQ=5$
By Appollonius theorem,
${AB}^2+{AC}^2=2{AQ}^2+2{BQ}^2$
$122=2{AQ}^2+2{\left(5\right)}^2$
$122=2{AQ}^2+2\times 25$
$122=2{AQ}^2+50$
$122-50=2{AQ}^2$
$72=2{AQ}^2$
$\frac{72}{2}={AQ}^2$
$36={AQ}^2$
$AQ=6$
$\therefore$ The length of the median is $6$ units.