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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 5, 2024
 #2
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Since M is the midpoint of QR​, then QM=MR=2QR​=26​=3.

 

Triangle PQR is a right triangle since ∠PQR=90∘ (by the Pythagorean Theorem, PQ2+QR2=PR2, so 52+62=72).

 

Since M is the midpoint of the hypotenuse of this right triangle, then segment PM is half the length of the hypotenuse. Therefore, PM = PR/2 = 7/2.

 May 7, 2024

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