Question

In a $\Delta ABC,AB=10cm,BC=12cm,andAC=14cm.$ Find the length of median AD. If G is the centroid, find length of GA :

• A. $\frac{5}{3}\sqrt{7},\frac{5}{9}\sqrt{7}$
• B. $5\sqrt{7},4\sqrt{7}$
• C. $\frac{10}{\sqrt{3}},\frac{8}{3}\sqrt{7}$
• D. $4\sqrt{7},\frac{8}{3}\sqrt{7}$

Answer 1

The correct option is D $4\sqrt{7},\frac{8}{3}\sqrt{7}$

By Apollonius theorem
$A{B}^2+A{C}^2=2(A{D}^2+B{D}^2)$ (BD = CD)
$\Rightarrow A{D}^2=112$
$\therefore AD=4\sqrt{7}$
Now, since $\frac{AG}{GD}=\frac{2}{1}$
$\Rightarrow AG=\frac{2}{3}AD$
$\therefore AG=\frac{2}{3}\times 4\sqrt{7}=\frac{8}{3}\sqrt{7}cm$