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#