Question

In $△PQR$, point $S$ is the midpoint of side $QR$. If $PQ=11$, $PR=17,PS=13$, find $QR$

Answer 1

In $△PQR$, point $S$ is the midpoint of side $QR$. [Given]
$\therefore$ seg PS is the median.
$\therefore {PQ}^2+{PR}^2=2{PS}^2+2{SR}^2[$ Apollonius theorem]
$\therefore {11}^2+{17}^2=2(13{)}^2+2{SR}^2$
$\therefore {11}^2+{17}^2=2(13{)}^2+2{SR}^2$ $\therefore 410+289=2\left(169\right)+2{SR}^2$
$\therefore 410=338+2{SR}^2$
$\therefore 2{SR}^2=410-338$
$\therefore 2{SR}^2=72$
$\therefore {SR}^2=72/2=36$
$\therefore SR=\sqrt{36}$ [Taking square root of both sides]
$=6$ units $NoW,QR=2SR[S$ is the midpoint of $QR]=2\times 6$
$\therefore QR=12$ units