Question

In $ABC$, the lengths of sides $BC,CA$ and $AB$ are $a,b$ and $c$ respectively. Median $AD$ drawn from $A$ is perpendicular to $AB$, then $b$ is:

Answer 1

Using Apollonius' Theorem:

The sum of the squares of any two sides of any triangle equals twice the square of half the third side together with twice the square on the median bisecting the third side.

Therefore, $b^2+c^2=2(\frac{a^2}{4}+A{D}^2)$

$\Rightarrow b^2+c^2=2(\frac{a^2}{4}+\frac{a^2}{4}-c^2)$

$\Rightarrow b^2=a^2-3c^2$

$\Rightarrow b=\sqrt{a^2-3c^2}$