Question

The medians AD and BE of a triangle with vertices $A(0,b),B(0,0)$ and $C(a,0)$ are perpendicular to each other, if

Answer 1

A median in a triangle is a line passing thought a vertex and through the midpoint of the side opposite to the vertex.

$D$ is the midpoint of side $BC$

$D=(\frac{a}{2},0)$

slope of $AD=\frac{b-0}{0-a/2}=\frac{-2b}{a}$

$E$ is the midpoint of side $AC$

$D=(\frac{a}{2},\frac{b}{2})$

slope of $BE=\frac{0-b/2}{0-a/2}=\frac{b}{a}$

If two lines are perpendicular, product of their slopes is -1.

$\left(\frac{-2b}{a}\right)\left(\frac{b}{a}\right)=-1$

$2b^2=a^2$

$a=\pm b\sqrt{2}$