Question

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.

Answer 1

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`