Question

One of the vertices of an equilateral triangle is $(2,3)$ and the equation of its opposite side is $x+y-2=0$. The area of triangle is

• A. $3 \sqrt{3}$
• B. $6 \sqrt{3}$
• C. $\frac{3 \sqrt{3}}{2}$
• D. $\sqrt{6}$

Answer 1

Answer: (C)

$AM =\frac{|2+3-2|}{\sqrt{1^{2}+1^{2}}}=\frac{3}{\sqrt{2}}$
$\tan 60^{\circ}=\frac{ AM }{ BM } $
$\Rightarrow BM =\frac{ AM }{\sqrt{3}}=\frac{3}{\sqrt{3} \cdot \sqrt{2}}=\frac{\sqrt{3}}{\sqrt{2}}$
$BC =2 BM =\sqrt{6}$

$\therefore $ Area of equilateral triangle $=\frac{\sqrt{3}}{4}(\sqrt{6})^{2}$
$=\frac{6 \sqrt{3}}{4}=\frac{3 \sqrt{3}}{2}$