Question

In triangle $PQR$, $PQ=13,QR=14,andPR=15$.

Let $\mathrm{M}$ be the midpoint of $\mathrm{QR}$.

Find $\mathrm{PM}.$

Answer 1

Explanation to correct answer:

In $∆PQR$

It is given that $M$ is the midpoint of the $\mathrm{QR}$

Therefore, $\mathrm{PM}$ is the median of the triangle.

By Apollonius theorem

$$\begin{gathered} {\mathrm{PQ}}^2+{\mathrm{PR}}^2=2({\mathrm{MR}}^2+{\mathrm{PM}}^2) \\ \Rightarrow {13}^2+{15}^2=2{\left(\frac{1}{2}\mathrm{QR}\right)}^2+2{\mathrm{PM}}^2 \\ \Rightarrow 169+225=\frac{49}{2}+2{\mathrm{PM}}^2 \\ \Rightarrow 2{\mathrm{PM}}^2=369.5 \\ \Rightarrow {\mathrm{PM}}^2=184.75 \\ \Rightarrow \mathrm{PM}=13.59 \end{gathered}$$

Hence, $\mathrm{PM}=13.59$