What do you think of this explanation for why perpendicualar lines have negative reciprocal slope?

This is something taught in high school math that is usually required to be accepted on faith. Here is my intuitive explanation, using the following diagram (it might be somewhat better as a video presentation).

Start with the point A = (5,2). Imagine the x and y axes as wires soldered together. Rotate the x and y axes so the x axis becomes the x’ axis going through A. The slope of the x’ axis is 5/2.

The y axis rotates to become the y’ axis. Since the y axis rotates through the same angle as the x axis, the angle between the y’ axis and the y axis is the same as the angle between the x and x’ axis. If we move down 5 and over 2, the point B that we get should be on the y’ axis. The slope of the y’ axis is therefore -5/2 and is of course perpendicular to the x’ axis.