In triangle PQR, by the law of cosines we have
[tex]34^2=32^2+34^2-2\cdot32\cdot34\cos(m\angle PQR)\implies\cos(m\angle PQR)=\dfrac8{17}[/tex]
In triangle PMQ, we have
[tex]PM^2=32^2+17^2-2\cdot32\cdot17\cos(m\angle PQR)\implies PM^2=801\implies PM=\boxed{3\sqrt{89}}[/tex]
