# State whether the following statement is true or false and justify : The vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is $x + y = 2$. Then the other two sides are $y - 3 = (2 \pm \sqrt3 ) (x - 2)$.

Toolbox:
• Slope of a line $m$ which make an angle $' \theta '$ in the positive direction of x axis is $m = \tan \theta$.
• Equation of a line with slope $m$ and passing through $(x_1, y_1)$ is $y-y_1=m(x-x_1)$
Step 1 :
Let $A(2,3)$ be one vertex and $x+y=2$ be the opposite side of the given equilateral triangle.
Hence the other two sides pass through the point A(2,3) and make an angle $60^{\circ}$
Slope of the line $x+y=2$ is 1.
$m = -1$
$\therefore$ Equation of the other two sides are
$y-3 = \bigg( \large\frac{-1- \tan 60^{\circ}}{1-\tan 60^{\circ}} \bigg) $$(x-2) and y-3 = \bigg( \large\frac{-1-\tan 60^{\circ}}{1+ \tan 60^{\circ}} \bigg)$$(x-2)$
But $\tan 60^{\circ} = \sqrt 3$
$\therefore y-3 = \large\frac{-1-\sqrt 3}{1-\sqrt 3 }$$(x-2) (i.e) y-3=\large\frac{-(1+ \sqrt 3 )}{1-\sqrt 3}$$(x-2)$
(i.e) $(y-3) = (2+\sqrt 3)(x-2)$
is the equation of one side.
$y-3 = \large\frac{\sqrt 3 - 1 }{\sqrt 3 + 1 }$$(x-2)$
$\Rightarrow (y-3)(2+ \sqrt 3 )=(x-2)$ is the equation of the other side.
(i.e) $(y-3)=(2-\sqrt 3 )(x-2)$
Hence the statement is true.