3: Equilateral Triangle


In this diagram, we have taken a simple square and on each side drawn an equilateral triangle each and our purpose is to find the coordinates of the vertices of these triangles. 

The only thing we need to know here is that the height of an equilateral triangle is aSin60 where a is the side of the triangles. We also find the equation of the line PQ. Note the circle with P, Q, R & S and the centre C taken out of the main diagram. 
Can you find the equations of the remaining lines like PS, QR & SR not to mention the simple ones like SQ & PR


There are two things to notice in this question.

The two lines are parallel since if you divide the second equation by 3 you  get y+2x=2 [mistakenly the diagram shows y+2x=3: go with it because the rest of the solution is very instructive]

Secondly, you must know how to compute the distance between two parallel lines.

And as per the question, the perpendicular distance between these two parallel lines happens to be the height of the equilateral triangle and thus, the result.