Question

The in centre of the triangle with vertices $(1,\sqrt{3}),(0,0)$ and $(2,0)$ is:

• A. $\left(1,\frac{\sqrt{3}}{2}\right)$
• B. $\left(\frac{2}{3},\frac{1}{\sqrt{3}}\right)$
• C. $\left(\frac{2}{3},\frac{\sqrt{3}}{2}\right)$
• D. $\left(1,\frac{1}{\sqrt{3}}\right)$

Answer 1

Answer: (D)

If triangle is an equilateral, then orthocentre, incentre and centroid will be same. Let $A(1,\sqrt{3}),B(0,0),C(0,1)$
be the given points.
$\therefore$ $a=BC$ $=\sqrt{{(2-0)}^2+{(0-0)}^2}=2$
$\therefore$ $b=CA$ $=\sqrt{{(2-1)}^2+{(0-\sqrt{3})}^2}=2$
$c=AB$ $=\sqrt{{(0-1)}^2+{(0-\sqrt{3})}^2}=2$
Triangle is equilateral.
$\therefore$ In centre is the same as centroid of the triangle.
$\therefore$ In centre is
$\left(\frac{1+0+2}{3},\frac{\sqrt{3}+0+0}{3}\right)$ $i.e.,$
$\left(1,\frac{1}{\sqrt{3}}\right)$