Question

In ∆ABC, point M is the midpoint of side BC.
If, AB² + AC² = 290 cm², AM = 8 cm, find BC.

Answer 1

In ∆ABC, point M is the midpoint of side BC.

$\mathrm{BM}=\mathrm{MC}=\frac{1}{2}\mathrm{BC}$

$$\begin{gathered} {\mathrm{AB}}^2+{\mathrm{AC}}^2=2{\mathrm{AM}}^2+2{\mathrm{BM}}^2\left(\mathrm{by}\mathrm{Apollonius}\mathrm{theorem}\right) \\ \Rightarrow 290=2{\left(8\right)}^2+2{\mathrm{BM}}^2 \\ \Rightarrow 290=2\left(64\right)+2{\mathrm{BM}}^2 \\ \Rightarrow 290=128+2{\mathrm{BM}}^2 \\ \Rightarrow 2{\mathrm{BM}}^2=290-128 \\ \Rightarrow 2{\mathrm{BM}}^2=162 \\ \Rightarrow {\mathrm{BM}}^2=81 \\ \Rightarrow \mathrm{BM}=9 \\ \therefore \mathrm{BC}=2\times \mathrm{BM} \\ =2\times 9 \\ =18\mathrm{cm} \end{gathered}$$

Hence, BC = 18 cm.