Question

The median to the base of an isosceles triangle is

• A. Inclined to ${60}^0$
• B. Inclined to ${30}^0$
• C. Inclined to ${45}^0$
• D. Perpendicular

Answer 1

The correct option is D Perpendicular

Let position vectors of $A,B,C$ be $\overline{o},\overline{b},\overline{c}$ respectively
$\overrightarrow{AD}$ is median
So, position vector of $\overline{d}=\frac{\overline{b}+\overline{c}}{2}$ (mid point on BC)
$|\overline{b}|=|\overline{c}|$ (Isosceles $\Delta$)
$\overrightarrow{AD}\cdot \overrightarrow{BC}=\left(\frac{\overline{b}+\overline{c}}{2}\right)\cdot (\overline{c}-\overline{b})$
$=\frac{|\overline{c}{|}^2-|\overline{b}{|}^2}{2}$
$=0$
So, $\overrightarrow{AD}\perp \overrightarrow{BC}$