Question

If the orthocentre of the triangle with vertices $\left(2, \frac{\sqrt{3}-1}{2}\right),\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\left(2,-\frac{1}{2}\right)$ is $(k, m)$, then evaluate $|k m|$.

Answer 1

$A =\left(2, \frac{\sqrt{3}-1}{2}\right), B =\left(\frac{1}{2},-\frac{1}{2}\right), C =\left(2,-\frac{1}{2}\right)$
Slope of $BC =\frac{-\frac{1}{2}+\frac{1}{2}}{2-\frac{1}{2}}=0$
$\Rightarrow BC \| x$ - axis.
Slope of $AC =\frac{\frac{\sqrt{3}-1}{2}+\frac{1}{2}}{2-2}$
$\Rightarrow AC \| y$ - axis
$\Rightarrow \angle C =90^{\circ}$
$\Rightarrow C$ is the orthocentre.
$\Rightarrow$ Orthocentre $=\left(2,-\frac{1}{2}\right)$
$\Rightarrow k=2, m=-\frac{1}{2}$
$\Rightarrow |k m|=1$