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In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$ [asy] unitsize(1 cm); pair P, Q, R; P = (1,3); Q = (0,0); R = (4,0); draw(P--Q--R--cycle); label("$P$", P, N); label("$Q$", Q, SW); label("$R$", R, SE); label("$5$", (P + Q)/2, NW, red); label("$7$", (P + R)/2, NE, red); label("$6$", (Q + R)/2, S, red); [/asy]

 May 6, 2024
 #1
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Since M is the midpoint of QR​, then QM=MR=2QR​=26​=3.

 

We now have a right triangle △PMQ with legs PQ=5 and QM=3. By the Pythagorean Theorem,

 

[PM^2 = PQ^2 + QM^2 = 5^2 + 3^2 = 25 + 9 = 34.]

 

Then, PM = sqrt(34).

 May 7, 2024

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