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

 Aug 11, 2024
 #1
avatar+1804 
+1

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! :)

 Aug 11, 2024
edited by NotThatSmart  Aug 11, 2024

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