Co-ordinates of incentre of the triangle formed having vertices (1, √3) (0, 0) and (2, 0) are

Question

Co-ordinates of incentre of the triangle formed having vertices \((1, \sqrt{3}),\) (0, 0) and (2, 0) are

(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

Correct option is (D) \(\left(1, \frac{1}{\sqrt{3}}\right)\)  

Let \(A(1, \sqrt3), \ B(0, 0), \ C(2, 0)\) be the vertices of a triangle ABC

\(a = BC = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)   

\(= \sqrt{(2-0)^2 + (0-0)^2} \)

\(= \sqrt{(2)^2}\)

= 2

\(b = AC = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)  

\(= \sqrt{(2-1)^2 + (0-\sqrt3)^2} \)  

\(= \sqrt{(1)^2 + (\sqrt3)^2}\)   

\(=\sqrt{1+3}\)   

\(=\sqrt{4}\)  

= 2

and \(c = AB = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)   

\(= \sqrt{(0-1)^2 + (0-\sqrt3)^2} \)    

\(= \sqrt{(1)^2 + (\sqrt3)^2}\)   

\(=\sqrt{1+3}\)   

\(=\sqrt{4}\)   

= 2

\(\therefore\) The triangle is an equilateral triangle.

\(\therefore\) Incentre is same as centroid of the triangle.

\(\Rightarrow\) Co - ordinates of incentre are 

\((\frac{1+0+2}{3}, \frac{\sqrt3+0+0}{3})\)

\(=(\frac{3}{3}, \frac{\sqrt3}{3})\)

\(=(1, \frac{1}{\sqrt3})\)