Question

In $\triangle pqr$, we have $pq = qr = 34$ and $pr = 32$. point $m$ is the midpoint of $\overline{qr}$. find $pm$.

Answer 1

In triangle PQR, by the law of cosines we have

$34^2=32^2+34^2-2\cdot32\cdot34\cos(m\angle PQR)\implies\cos(m\angle PQR)=\dfrac8{17}$

In triangle PMQ, we have

$PM^2=32^2+17^2-2\cdot32\cdot17\cos(m\angle PQR)\implies PM^2=801\implies PM=\boxed{3\sqrt{89}}$