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Q.
The in centre of the triangle with vertices $ (1,\,\,\sqrt{3}),\,\,(0,\,\,0) $ and $ (2,\,\,0) $ is:
Jharkhand CECEJharkhand CECE 2006
Solution:
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) $