Question

In the figure, seg $AQ$ is the median. ${AB}^2+{AC}^2=122$. $BC=10$. Find the length of the median $AQ$.

Answer 1

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