In triangle PQR, PQ=13, QR=14, and PR=15. Let M be the midpoint of QR. Find PM.
+26367 In triangle PQR, PQ=13, QR=14, and PR=15.
Let M be the midpoint of QR.
Find PM.
cos Rule:
\(\begin{array}{|rcll|} \hline 15^2&=&13^2+14^2-2*13*14*\cos(Q) \\ 2*13*14*\cos(Q) &=& 13^2+14^2-15^2 \\ 2*13*14*\cos(Q) &=& 140 \quad | \quad : 2 \\ 13*14*\cos(Q) &=& 70 \qquad (1) \\ \hline \end{array}\)
cos Rule:
\(\begin{array}{|rcll|} \hline PM^2 &=& 13^2+\left(\dfrac{14}{2}\right)^2-2*13*\dfrac{14}{2}*\cos(Q) \\\\ PM^2 &=& 13^2+\left(\dfrac{14}{2}\right)^2-13*14*\cos(Q) \\\\ 13*14*\cos(Q) &=& 13^2+\left(\dfrac{14}{2}\right)^2-PM^2 \\\\ 13*14*\cos(Q) &=& 13^2+7^2-PM^2 \\\\ 13*14*\cos(Q) &=& 218-PM^2\qquad (2) \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline (1)=(2):& 70 &=& 218-PM^2 \\ & PM^2&=& 218 - 70 \\ & PM^2&=& 148 \\ & PM^2&=& 4*37 \\ & \mathbf{PM}^2 &=&\mathbf{ 2\sqrt{37} } \\ \hline \end{array}\)
