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  4. The medians A D and B E of the triangle with vertices A(0, b), B(0,0) and C(a, 0) are mutually perpendicular if

Q. The medians $A D$ and $B E$ of the triangle with vertices $A(0, b), B(0,0)$ and $C(a, 0)$ are mutually perpendicular if

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Solution:

Here $D \equiv(a / 2,0)$ and $E \equiv(a / 2, b / 2)$.
Now slopes of $AD$ and $BE$ are $\frac{-b}{ a / 2}$ and $\frac{ b / 2}{ a / 2}$.
Product of the slopes $=-1 $
$\Rightarrow \frac{-b^{2} / 2}{a^{2} / 4}=-1$
$ \Rightarrow a^{2}=2 b^{2} $
$\Rightarrow a=\pm \sqrt{2} b$.

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