M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN=12, and PR=14, then find OR.
Thank you in advance
+9519 Obviously. O is the centroid of \(\triangle PQR\).
Because PN = NR, and QN is perpendicular to PR, PQ = QR.
And,
\(PQ^2 = PN^2 + NQ^2 = \left(\dfrac12 \cdot 14\right)^2 + 12^2 = 193\\ PQ = \sqrt{193} = QR\)
Using Apollonius's theorem,
\(MR = \dfrac12 \sqrt{2\cdot 14^2 + 2\cdot \sqrt{193}^2 - \sqrt{193}^2} = \dfrac32 \sqrt{65}\)
Because centroid divides each medians into 2 : 1,
\(OR = \dfrac23 MR = \boxed{\sqrt{65}}\)
