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 = 18, then find OR.

Guest Jun 15, 2022

#2**+1 **

By Menelaus' theorem,

\(\dfrac{QO}{ON} \cdot \dfrac{NR}{RP}\cdot \dfrac{PM}{MQ} = 1\\ \dfrac{QO}{ON} \cdot \dfrac12 \cdot \dfrac11 = 1\\ \dfrac{QO}{ON} = 2\\ QO = 2 ON\)

Since QN = 12, QO = 8 and ON = 4.

Since PN = NR and PR = 18, NR = 9.

Then by Pythagorean theorem,

\(OR = \sqrt{NR^2 + ON^2} = \sqrt{4^2 + 9^2} = \sqrt{97}\)

MaxWong Jun 15, 2022