Question

If one side of an equiateral triangle is $3x+4y=7$ and its vertex is $(1,2)$, then the length of the side of the triangle is

• A. $\frac{4\sqrt{3}}{17}$
• B. $\frac{3\sqrt{3}}{16}$
• C. $\frac{8\sqrt{3}}{15}$
• D. $\frac{4\sqrt{3}}{15}$

Answer 1

The correct option is C $\frac{8\sqrt{3}}{15}$

Let $d$ be the perpendicular distance of $C$ from line $AB$

So, $d=\left|\frac{3+8-7}{\sqrt{25}}\right|=\left|\frac{4}{5}\right|$

$d=\frac{4}{5}$ (since, $d>0$)

So, from figure, $a\cos {30}^{\circ }=\frac{4}{5}$

$\Rightarrow a=\frac{8}{5\sqrt{3}}=\frac{8\sqrt{3}}{15}$