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.
+130071 Draw MJ perpendicular to PR to meet PR at J
Draw MK parallel to PR to meet QN at K
MK divides QN into equal parts so KN = 6 = MJ
And
PJ / MJ = PN /NQ
PJ/ 6 = 9/12 = 3/4
PJ = 18/4 = 9/2 = 4.5
So RJ = 18 - 4.5 = 13.5
And
ON / NR = MJ / RJ
ON / 9 = 6 / 13.5
ON = 54/13.5 = 4
So using the Pythagorean Theorem:
OR = sqrt ( NR^2 + ON^2) = sqrt ( 4^2 + 9^2) = sqrt (97)
