Math 150


Dr. Wilson

The Feet in the Mirror Problem

Suppose you are getting ready to go somewhere where you want to look nice, so, before you leave, you check to see how you look in the mirror. You are also interested in seeing how well your shoes go with the rest of your outfit, but you can't see your feet in the mirror. The reflection in the mirror goes only to your knees. Do you move closer to the mirror to be able to see your feet or do you move farther away?

It turns out that if you are standing straight up and the mirror is also vertical, it won't make any difference. No matter whether you move closer or farther away, you will always be able to see to the same point on your legs. Consider the following diagram

T is the top of the mirror, B is the bottom of the mirror, E is the location of your eye, and L is as far down on your legs as you can see. F is your feet, and W is the point on your waist which is at the same level as the bottom of the mirror. The law of physics that governs reflections is called Snell's Law which says that the angle of incidence, /TBE is the same size as the angle of reflection, /ABL. From this we can extract the following given information.

This says that the distance from your eye to how far down you can see on your leg will always be twice the distance from the level of you eye to the level of the bottom of the mirror, no matter where you stand.

If we examine out assumptions, there is nothing we can do about the one where BW || AB. This comes from the definition of W as the point on your waist which is the same height as the botom of the mirror. But the ones where we have AT and EF perpendicular to AF can be changed. They say that the mirror is plumb to the floor and that you are standing up straight. These can be adjusted, and you will see that mirrors which are designed to enable you to see your feet are often adjustable so that you can move the top away from the wall. Another way to defeat this proof would be to stand close to the mirror and lean over.

For a proof of Snell's Law click here.