Question

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

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

Answer 1

Answer: (D)

Key Idea : If the triangle is equilateral, then incentre is coincide with centroid of the triangle.
Let $ A(1, \sqrt{3}) B(0,0), C(2,0)$ be the vertices of a triangle $A B C$.
$a=B C=\sqrt{(2-0)^{2}+(0-0)^{2}}=2 $
$b=A C=\sqrt{(2-1)^{2}+(0-\sqrt{3})^{2}}=2$
and $ c=A B=\sqrt{(0-1)^{2}+(0-\sqrt{3})^{2}}=2$
$\therefore $ The triangle is an equilateral triangle.
$\therefore $ Incentre is same as centroid of the triangle.
$\Rightarrow $ Co-ordinates of incentre are
$\left(\frac{1+0+2}{3}, \frac{\sqrt{3}+0+0}{3}\right) \text { i.e., }\left(1, \frac{1}{\sqrt{3}}\right)$