# If an equilateral triangle, having centroid at the origin, has a side along the line, x + y = 2, then the area (in sq. units) of thistriangle is : Option 1) Option 2) 6 Option 3) Option 4)

As learnt in

Perpendicular distance of a point from a line -

$\rho =\frac{\left | ax_{1}+by_{1}+c\right |}{\sqrt{a^{2}+b^{2}}}$

- wherein

P  is the distance from the line $ax+by+c=0$ .

$AD\:=\:P\:=\frac{(2-1-2)}{\sqrt{1^{2}+1^{2}}}\\=\frac{1}{\sqrt{2}}$

$\frac{P}{a}=sin60^{\circ}=> \:a=\frac{P}{sin60^{\circ}}\:=\frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{6}}=\frac{\sqrt{2}}{3}$

Area = $\frac{\sqrt{3}}{4}\:\left (\frac{\sqrt{2}}{3} \right )^{2}$

= $6\sqrt{3}$

Option 1)

This option is incorrect.

Option 2)

6

This option is incorrect.

Option 3)

This option is correct.

Option 4)

This option is incorrect.

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