Mirror on the Wall

Assume you are 66'' tall and looking at a wall mirror 5' away. How long the mirror should be so that you'd see both your toes and the head?

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Explanation

Perhaps curiously, the size of the mirror does not depend on a few parameters that one might intuit could be relevant. The distance to the mirror or the relative height of your eyes do no affect the mirror size. The only relevant parameter is your height. The mirror must be half of that.

To see that, consider phantom images H, E, T of your head, eyes and toes in the mirror as you perceive them. These are at the same height and the distance from the mirror but on the other side of the wall. Let A be the point where the lines EH' and E'H meet and C is defined as the intersection of ET' and E'T. AC is a midline of triangle E'HT and thus equals HT/2. AC is the shortest mirror in which you can see your whole figure.

References

1. D. Rohrer, More Thought Provokers, Key Curriculum Press, 1994, #21