Answer: QR = 12
S is the midpoint of QR, so PS is a median of \(\Delta PQR\). By Apollonius' theorem,
\(PQ^2 + PR^2 = 2PS^2 + 2QS^2\)
\(11^2 + 17^2 = 2(13)^2 + 2QS^2\)
\(121 + 289 = 338 + 2QS^2\), i.e. \(410 = 338 + 2QS^2\)
\(2QS^2 = 72\), so \(QS^2 = 36\) and \(QS = 6\).
\(\therefore QR = 2\,QS = 2 \times 6 = 12\).