Question

In ∆ABC, AB = 10, AC = 7, BC = 9 then find the length of the median drawn from point C to side AB

Answer 1

In ∆ACB, point D is the midpoint of side AB.

$\mathrm{AD}=\mathrm{BD}=\frac{1}{2}\mathrm{AB}=5$

$$\begin{gathered} {\mathrm{CA}}^2+{\mathrm{CB}}^2=2{\mathrm{DC}}^2+2{\mathrm{AD}}^2\left(\mathrm{by}\mathrm{Apollonius}\mathrm{theorem}\right) \\ \Rightarrow 7^2+9^2=2{\mathrm{DC}}^2+2\left(5^2\right) \\ \Rightarrow 49+81=2{\mathrm{DC}}^2+2\left(25\right) \\ \Rightarrow 130=2{\mathrm{DC}}^2+50 \\ \Rightarrow 2{\mathrm{DC}}^2=130-50 \\ \Rightarrow 2{\mathrm{DC}}^2=80 \\ \Rightarrow {\mathrm{DC}}^2=40 \\ \Rightarrow \mathrm{DC}=2\sqrt{10} \end{gathered}$$

Hence, the length of the median drawn from point C to side AB is $2\sqrt{10}$.