+697 In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11
+130099 P
5 8
Q M R
5.5 5.5
Note that cos PMR = -cos PMQ
Law of Cosines
PQ^2 = PM^2 + QM^2 - 2 ( PM * QM) cos (PMQ)
PR^2^2 = PM^2 + RM^2 - 2(PM * RM) cos (PMR)
5^2 = PM^2 + 5.5^2 - 2 (PM * QM) cos (PMQ)
8^2 = PM^2 + 5.5^2 - 2(PM * 5.5) (- cos PMQ)
5^2 = PM^2 + 5.5^2 - 2 (PM * QM) cos (PMQ)
8^2 = PM^2 + 5.5^2 + 2(PM * 5.5) ( cos PMQ) add these
89 = 2PM^2 + 60.5
[89 - 60.5] / 2 = PM^2
PM = sqrt [ 28.5 / 2 ] ≈ 3.775
