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 $|km|$.

Answer 1

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