The midpoint of PQ is M. The midpoint of PR is N. QN and RM intersect at O. If QN is perpendicular to PR, QN = 10, and PR = 16, then find OR.
+421 If we draw QR, as QN is a perpendicular and midpoint, QPR is isosceles with QP=QR. This means that O is the centroid of QPR, so OR = 2*OM. However, we are not sure how to find OM, but see that we can draw a perpendicular MX from M to PN, which gives us similar triangles PMX and PQN. Next, we note that as PN = 16/2 = 8, and QN = 10, PQ = 2√41. Thus, PM = √41.
As PM/PQ = MX/QN = PX/PN, X is the midpoint of PN, and MX = 5. Thus, PX = 4. So, as MX = 5 and XR = 16 - 4 = 12, MR = 13, so OR = 13/3 * 2 = 26/3.
+421 If we draw QR, as QN is a perpendicular and midpoint, QPR is isosceles with QP=QR. This means that O is the centroid of QPR, so OR = 2*OM. However, we are not sure how to find OM, but see that we can draw a perpendicular MX from M to PN, which gives us similar triangles PMX and PQN. Next, we note that as PN = 16/2 = 8, and QN = 10, PQ = 2√41. Thus, PM = √41.
As PM/PQ = MX/QN = PX/PN, X is the midpoint of PN, and MX = 5. Thus, PX = 4. So, as MX = 5 and XR = 16 - 4 = 12, MR = 13, so OR = 13/3 * 2 = 26/3.
