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 $ABC$.
$a=BC=\sqrt{(2-0{)}^2+(0-0{)}^2}=2$
$b=AC=\sqrt{(2-1{)}^2+(0-\sqrt{3}{)}^2}=2$
and $c=AB=\sqrt{(0-1{)}^2+(0-\sqrt{3}{)}^2}=2$
$\therefore$ The triangle is an equilateral triangle.
$\therefore$ Incentre is same as centroid of the triangle.
$\Rightarrow$ Co-ordinates of incentre are
$\left(\frac{1+0+2}{3},\frac{\sqrt{3}+0+0}{3}\right)\text{ i.e., }\left(1,\frac{1}{\sqrt{3}}\right)$