question_answer

The vertex of an equilateral triangle is \[(2,\,\,-1)\] and the equation of its base is \[x+2y=1\], the length of its sides is:

A) \[2/\sqrt{15}\]                 

B) \[4/3\sqrt{3}\]

C) \[1/\sqrt{5}\]                    

D) \[4/\sqrt{15}\]

Correct Answer: A

Solution :

Key Idea: In equilateral triangle all sides of a triangle are equal. Given equation of \[BC\] is\[x+2y=1\] The perpendicular distance from a point\[A(2,\,\,-1)\]is                 \[AD=\frac{|2+2(-1)-1|}{\sqrt{1+4}}=\frac{|-1|}{\sqrt{5}}\]                        \[=\frac{1}{\sqrt{5}}\] In\[\Delta ABD\],                 \[\sin {{60}^{o}}=\frac{AD}{AB}\] \[\Rightarrow \]               \[AB=\frac{1}{\sqrt{5}}\times \frac{2}{\sqrt{3}}\]                       \[=\frac{2}{\sqrt{15}}\] Note: In equilateral triangle, median of a triangle is the mid poin.t of base.