Question

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

• A. $\sqrt{\frac{20}{3}}$ units
• B. $\frac{2}{\sqrt{15}}$ units
• C. $\sqrt{\frac{8}{15}}$ units
• D. $\sqrt{\frac{15}{2}}$ units

Answer 1

Answer: (A)

$\Delta ABC$ is an equilateral triangle. $AD$ is altitude.Equation of $BC$ is $2x-y=1$



So perpendicular distance,
$AD=\left|\frac{2\left(-1\right)-2-1}{\sqrt{4+1}}\right|$
$=\left|-\sqrt{5}\right|=\sqrt{5}$
In $\Delta ABD$, $\frac{AD}{AB}=sin60^{\circ }$
$\Rightarrow \frac{\sqrt{5}}{AB}=\frac{\sqrt{3}}{2}$
$\therefore AB=\frac{2\sqrt{5}}{\sqrt{3}}=\sqrt{\frac{20}{3}}$ units