Question

A vertex of an equilateral triangle is $(0,0)$ and the equation of the opposite side is $x+y=2$ . Find the equations of the other two sides and the length of each side of the triangle.

Answer 1

$OAB$ is equilateral, $PQ:x+y=2$

$\angle AOB={60}^o$

$\angle QOP={90}^o$

$\Rightarrow \angle QOA={15}^o,\angle BOP={15}^o$

Equation of $OB:y=\tan {15}^{o}x=(2-\sqrt{3})x$

Equation of $OA:y=\tan {75}^{o}x=(2-\sqrt{3})x$

$\perp$ distance of $O$ to $AB=\frac{|0to-2|}{\sqrt{2}}=\frac{3\sqrt{3}a}{2}$

$\Rightarrow a=\frac{2\sqrt{2}}{\sqrt{3}}\to$ side length