Question

One vertex of an equilateral triangle is $(2,3)$ and the equation of the side is $x-y+5=0$ then find the area of the equilateral triangle.

Answer 1

Since the point $(2,3)$ does not satisfy the line $x-y+5=0$.......(1)

So the line (1) is the opposite side to the vertex $(2,3)$.

Now the perpendicular distance of $(2,3)$ from the line (1) is $\left|\frac{2-3+5}{\sqrt{1^2+(-1{)}^2}}\right|=\frac{4}{\sqrt{2}}=2\sqrt{2}$.

This perpendicular distance is nothing but the height of the equilateral triangle.

Then $\frac{\sqrt{3}}{2}\times$ side $=2\sqrt{2}$

or, side $=\frac{4\sqrt{2}}{\sqrt{3}}$.

Now area of the equilateral triangle is $\frac{\sqrt{3}}{4}\times$ side ${}^2=\frac{8}{\sqrt{3}}$ sq. units.