What is the length between PO, PQ, OQ have to be when the points form a equilateral triangle?

There are 2 lines on a cartesian axis. One line is defined by Y=-2X+1 and point O and Q are an element of it. The other line is described as y=-2X+2 and point P is an element of it. I would appreciate it if you could explain it step by step with the use formula's.

1 Answer
Jul 18, 2018

There are 2 lines on a cartesian axis. One line is defined by Y=-2X+1 and point O and Q are an element of it. The other line is described as y=-2X+2 and point P is an element of it. I would appreciate it if you could explain it step by step with the use formula's.

Given that eqution of line-1on which the vertices O and Q of the equilateral triangle POQ lie, is
y=-2x+1or2x+y=1

It is also given that the vertex P of the triangle lies on the line-2 having eqution y=-2x+2or2x+y=2 These two lines are parallel as their slopes are same.So distance between them must be height (h) of the equilateral DeltaPOQ

The normal forms of these lines are

"Line"_1->2/sqrt5x+1/sqrt5y=1/sqrt5

And

"Line"_2->2/sqrt5x+1/sqrt5y=2/sqrt5

Hence the perpendicular distance of these parallel straight linse or the height of the equilateral DeltaPOQ will be h=2/sqrt5-1/sqrt5=1/sqrt5

If PO=OQ=PQ=a then height of DeltaPOQ

h=sqrt3/2a

So

sqrt3/2a=1/sqrt5

Hence
PO=OQ=PQ=a=2/sqrt15