Two dots on toilet paper roll problem

Question

I have an empty toilet paper roll.

-I grab a pencil and draw two dots on the outside of the roll in random places.

-I connect these dots with a straight line, along the shortest path.

-I grab some scissors, and cut the roll vertically (where the holes are on the top and bottom of the roll). The cut is perfectly straight, through a random point on the line.

The line is now severed and I am left with a rectangular piece of cardboard with two dots and now TWO lines instead of one. One line going from dot 1 to its closest edge, and the other going from dot 2 to its closest edge.

If I erase both lines completely, how can I get them back using math?

I know the distance from each dot to their respective edge, as well as the distance from each dot to the top and/or bottom of the now flat, rectangular toilet paper roll.

I need to figure out how to draw a line from each dot, to a point on their edge of the rectangle, so that if I roll it back up into a cylinder again, the line is straight like it was in the beginning.

The distance from the top or bottom of the roll to where each line meets their respective edge should be exactly the same, as these lines were once a single line.

Answer 1

You have erased the lie and your figure looks like

Make 2 additional copies of your figure

and connect the dots.

And delete the left and right copies.

Answer 2

As the figure shows, you have the two points $A(x_1, y_1)$ and $B(x_2, y2)$, and you know length $L$.

You only need to find $h$ (the distance along the vertical from the bottom). The condition to be satisfied by the line is that the slope of the two segments is the same.

The first segments extends from $(0, h)$ to $(x_1, y_1)$, and the second segment from $(x_2, y_2)$ to $(L, h)$. Therefore, we have

$ \dfrac{ y_1 - h }{x_1} = \dfrac{ h - y_2 }{L - x_2} $

Cross multiplying,

$ (L - x_2) (y_1 - h) = x_1 (h - y_2) $

Therefore,

$ h = \dfrac{ x_1 y_2 + (L - x_2) y_1 }{ x_1 + L - x_2 } $

Now you can draw the two line segments.

Answer 3

Construction using parallel lines: