Question

The coordinates of the orthocentre of the triangle with vertices $\left(2,\frac{\sqrt{3}-1}{2}\right),\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\left(2,-\frac{1}{2}\right)$ are ________.

Answer 1

Given vertices are
$\left(2,\frac{\sqrt{3}-1}{2}\right),\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\left(2,-\frac{1}{2}\right)$


Here slope of BC = $\frac{-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}}{2-{\displaystyle \frac{1}{2}}}=0$
Slope of AC = $\frac{{\displaystyle \frac{\sqrt{3}-1}{2}}+{\displaystyle \frac{1}{2}}}{2-2}=\infty$
i.e Slope of BC is O and slope of AC is ∞
⇒ BC and AC are perpendicular to each other at angle C
i.e ∠C = 90^o
Since orthocentre of right angled triangle is at C point
∴ Co-ordinate of orthocentre at $\left(2,-\frac{1}{2}\right)$