Question

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

Answer 1

In $△PQR,$

$PQ=17units,PR=11units,QR=?,PS=13units$

We know that Apollonius's theorem relates the length of a median of a triangle to the lengths of its side. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".

Specifically, in any $△ABC$, $AD$ is a median, then

$A{B}^2+A{C}^2=2(B{D}^2+C{D}^2)$

Using Apollonius's theorem,

$P{Q}^2+P{R}^2=2(P{S}^2+S{R}^2)$

${17}^2+{11}^2=2({13}^2+S{R}^2)$

$289+121=2(169+S{R}^2)$

$410=2(169+S{R}^2)$

$(169+S{R}^2)=205$

$S{R}^2=36$

$SR=6$ cm

Since $S$ is the midpoint of $QR$

$QR=2\times SR$

$QR=12$

$QR=12$ cm.