Question

If the equation of the base of an equilateral triangle is $x+y=2$ and its vertex is $(2,-1)$ then find the length of the side of the triangle.

Answer 1

REF.Image

Let a be the length of each side of the equilateral triangle. The length perpendicular P from (2,-1) on x+y-2=0.

$\Rightarrow P=\left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right|=\left|\frac{2\times 1+1\times -1-2}{\sqrt{1^2+1^2}}\right|$

$=\left|\frac{2-1-2}{\sqrt{2}}\right|=\frac{1}{\sqrt{2}}$

Consider the $\Delta ABD$

sin${60}^{\circ }=p/a\Rightarrow \frac{\sqrt{3}}{2}=\frac{\frac{1}{\sqrt{2}}}{a}$

$\Rightarrow a=\frac{2}{\sqrt{3}}\times \frac{1}{\sqrt{2}}$

$=\frac{2}{\sqrt{6}}\times \frac{\sqrt{6}}{\sqrt{6}}$

$=\frac{2\sqrt{6}}{6}=\frac{\sqrt{6}}{3}$

Hence length of the side of the equilateral triangle is $\frac{\sqrt{6}}{3}$