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In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$

 

 Jun 18, 2024
 #1
avatar+129732 
+1

 

 

cos PMR  = -cos PMQ

 

Law of Cosines (twice)

 

PO^2  = QM^2  + PM^2 - 2(QM * PM) cos PMQ

PR^2  = RM^2 + PM^2 - 2(RM * PM) ) (-cos PMQ)

 

18^2  = 12.5^2 + PM^2 - 2(12.5 * PM) cos PMQ

23^2 = 12.5^2 + PM^2 + 2(12.5 * PM) cos PMQ     add these

 

18^2 + 23^2  =  2*12.5^2 + 2PM^2

 

853 = 312.5 + 2PM^2

 

540.5 = 2PM^2

 

270.25 = PM^2

 

sqrt [270.25] = PM ≈ 16.44

 

cool cool cool

 Jun 18, 2024

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