+1459 In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11
+1926 First, let's use the Law of Cosines. It states that \(cos PMQ = - cos PMR\)
Thus, we can write the equations
\(PQ^2 = QM^2 + PM^2 - 2(QM * PM) * (-cos PMR) \\ PR^2 = RM^2 + PM^2 - 2 (RM * PM) * ( cos PMR)\)
Plugging in the values we already know from the problem, we get
\(5^2 = 5.5^2 + PM^2 + 2(5.5 * PM)cos(PMR) \\ 8^2 = 5.5^2 + PM^2 - 2(5,5 * PM) cos (PMR)\)
Now, add these two equations. We get
\(5^2 + 8^2 = 2 * 5.5^2 + 2PM^2 \\ 89 = 60.5 + 2PM^2 \\ 28.5 / 2 = PM^2 \\ 14.25 = PM^2\)
Thus, we have \(\sqrt{14.25} = PM ≈ 3.77\)
The answer is 3.77.
Thanks! :)
