Which pair of points forms a vertical line segment that is 4 units long?

In a Cartesian coordinate system, a vertical line segment always is parallel to the y-axis, so we have this coordinate system in the figure below. We need to find a pair of points that is 4 units long.

So, this pair of points are in the general forms as follow:

[tex]P_{1}( x_{1}, y_{1})[/tex]
[tex]P_{2}( x_{2}, y_{2})[/tex]

Given that the straight line is vertical, then:

[tex]x_{1}=x_{2}[/tex]
 
Then the point 2 is modified as:

[tex]P_{2}( x_{1}, y_{2})[/tex]

We need to find a point which is 4 units long, so:

[tex]d = y_{2} - y_{1} = 4[/tex]
∴ [tex]y_{2} = 4+ y_{1}[/tex]

So, we can find infinite points given a value of [tex]y_{1}[/tex], therefore if:

[tex]y_{1} = 2[/tex] then:

∴ [tex]y_{2} = 6[/tex]

Thus, if [tex] x_{1} = 1 [/tex] then a pair of points are:

[tex]P_{1}(1,2)[/tex]
[tex]P_{2}(1,6)[/tex]