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)

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

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