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In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

 Mar 20, 2024
 #1
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          P

      5            8

 

Q         M            R

            11

 

Law of Cosines

PR^2  = QP^2 + QR^2  - 2(QP * QR)cos(PQR)

8^2   = 5^2 + 11^2  - 2(5 * 11) cos (PQR)

64  = 25 + 121 - 110 cos (PQR)

[ 64 - 25 - 121 ]  / [ -110]  = 41/55 cos (PQR)

 

Law of Cosines again

PM^2  = QM^2  + PQ^2 - 2 (QM * PQ)cos (PQR)

PM^2  = (5.5)^2  + 5^2  - 2(6.5 * 5) (41/55)

PM^2  = 14.25

PM = sqrt [14.25  ]  ≈   3.77

 

 

cool cool cool

 Mar 20, 2024

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