Question

The equation of the base BC of an equilateral triangle ABC is $x+y=2$ and A is $(2,-1)$ . The length of the side of the triangle is:

• A. $\sqrt{2}$
• B. ${\left(\frac{3}{2}\right)}^{\frac{1}{2}}$
• C. ${\left(\frac{1}{2}\right)}^{\frac{1}{2}}$
• D. $\left(\frac{1}{\sqrt{2}}\right)$
• E. ${\left(\frac{2}{3}\right)}^{\frac{1}{2}}$

Answer 1

Answer: (E)

Length of perpendicular from $A(2,-1)$ to the line $x+y-2=0$ is $\left|\frac{2-1.2}{\sqrt{1+1}}\right|=\frac{1}{\sqrt{2}}$
In $\Delta ABC,$ $\frac{AD}{AB}=sin60{}^{\circ }$ $\Rightarrow$ $\frac{1}{\sqrt{2}AB}=\frac{\sqrt{3}}{2}$ $\Rightarrow$ $AB=\sqrt{\frac{2}{3}}$