UPSC NDA Math Model Paper 7

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Given, BE and AD are medians of the triangle with vertices A(0, b), B(0, 0) and C(a, 0). ⇒ D is the midpoint of the side AB and E is the midpoint of the side AC By mid point formula, we have The coordinate of the point $D=D=\left(\frac{a}{2}, 0\right)$ The coordinate of the point $E=E=\left(\frac{a}{2}, \frac{b}{2}\right)$ Slope of $A D=m_{1}=\frac{b-0}{0-\frac{a}{2}}=\frac{-2b}{a}$ Slope of $B E=m_{2}=\frac{\frac{b}{2}-0}{\frac{a}{2}-0}=\frac{b}{a}$ The medians of the triangle are perpendicular to each other and the product slope of perpendicular line is -1. $\Rightarrow m_1 \cdot m_2 = -1$ $\Rightarrow \frac{-2b}{a} \cdot \frac{b}{a} = -1$ $\Rightarrow a^{2}=2 b^{2}$ $\Rightarrow a = \pm \sqrt{2b}$ The median BE and AD of a triangle with vertices A(0, b), B(0, 0) and C(a, 0) are perpendicular to each other if $a = \pm \sqrt{2b}$