Why do these paths all have the same length?

In a high school book the author said these paths have all the same distance. Is that true? how to convince myself (and my students as well) they all have the same length?

Solution 1:

By a sequence of reflections of setions of each path ... each path can be shown to the same length as the red path.

Solution 2:

Turn the picture by about $55^\circ$ clockwise, so that the paths are made of horizontal and vertical segments.

Then you will see that any of those paths is as long as it takes to move from $A$ to the 'right', all the way to the place 'right above $B$' and then move 'down' all the way to $B$ (darker black line added on the picture).

Solution 3:

When I was 5th grade student or something like that, I couldn't understand how they could be equal in a similar question and one of my brothers showed me a figure as I drawed and I completely understand it when I saw that figure. The figure I attached is as same as that one and since I didn't want to mess up the original one, I showed it on path $I$ and $II$:

Note that same colored segments have the same length.

Solution 4:

Translating vectors doesn't change their magnitude.

Or:

The small red (green) vectors sum to the big red(green) vector.