+414 In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11
+130462 P
5 8
Q 5.5 M 5.5 R
Law of Cosines
Two Equations
5^2 = 5.5^2 + PM^2 - 2 ( 5.5 * PM) cos PMQ
8^2 = 5.5^2 + PM^2 - 2 (5.5 * PM) cos PMR
cos PMR = -cos PMQ
So
5^2 = 5.5^2 + PM^2 - 2 (5.5 * PM) cos PMQ
8^2 = 5.5^2 + PM^2 + 2(5.5* PM) cos PMQ add these
89 = 60.5 + 2PM^2
(89 -60.5) / 2 = PM^2
PM^2 = 14.25
PM =sqrt [14.25 ] ≈ 3.775
