Question

In the following figure in $\Delta ABC$, $AP$ is the median. If $AP=12,A{B}^2+A{C}^2=320$, then find $BP$.

Answer 1

In $\Delta ABC$
By Appollonius theorem,
$A{B}^2+A{C}^2=2A{P}^2+2B{P}^2$
$320=2(12{)}^2+2(BP{)}^2$
$320=2\times 144+2B{P}^2$
$320=288+2B{P}^2$
$2B{P}^2=320-288$
$2B{P}^2=320-288$
$2B{P}^2=32$
$B{P}^2=\frac{32}{2}$
$B{P}^2=16$
$BP=4$