# University of Wisconsin Green Bay

Your friend is moving into a new apartment. He has a 4.0 foot tall mirror that he wants to hang on his wall so that he can see himself fully in it. If your friend is six foot four, how high should he hang the mirror? Assume that his eyes are four inches below the top of his head.

• Mirrors reflect light. To understand what you see or how you see it when looking in a mirror, you need to think about how light is reflected.

• In order to understand what we see in a mirror, we need to draw the light rays that get to our eyes. In this case, your friend wants to see all of himself. As long as he can see his feet and the top of his head, the mirror is located well on the wall. Therefore, you need to draw the light ray that goes from his feet to his eyes, and the light ray that goes from the top of his head to his eyes.

• Beyond the understanding that Θi = Θr (used in the Draw a Picture stage), understanding the reflection off a plane mirror just requires geometry. At the most, you may need to call on the definitions of sine, cosine or tangent, but more likely you will compare similar triangles.

• Note that the two triangles outlined in red (one shaded and one clear) are similar triangles. Furthermore, they share a horizontal side so they are similar triangles of the same size. This means that the two vertical sides are the same length. Since they add to 6 ft, each vertical side is 3 feet. Your friend will be able to see his feet as long as the bottom of the mirror is no more than 3 feet off the floor.

You can repeat this analysis for the top of your friend's head. In that case, the two sides of the similar triangles add to 4 inches, so each side is 2 inches. Your friend will be able to see the top of his head as long as the top of the mirror is at least 6' 2" off the floor.

• Mirrors redirect light waves by reflecting them off the surface of the mirror. Your brain interprets the light that gets to your eye as if it traveled in a straight line and so your friend sees an image of himself that looks like it is coming from behind the mirror.

As long as the mirror is high enough to reflect a ray of light from your friend's head to his eye, he will see his head. As long as it is low enough to reflect a ray of light from his foot to his eye, he will see his foot. You know that light will travel in a straight line unless it comes to an interface, and so you are able to draw the paths that the light rays take between your friend and his eyes. It is only a matter of using geometry to relate the distances in your figure.

Now is also a good time to return to the first FAQ. Note that regardless of the distance between your friend and the mirror, the vertical sides of the lower triangles are equal and add to six feet, and the vertical sides of the upper triangles are equal and add to four inches. As long as the mirror is at least half your friend's height and hung correctly, he will be able to see himself no matter how far back he stands. If you feel that your experience is contrary to this result, pay very careful attention next time you look in a mirror. There is a distance effect for a mirror hung at a slant, but not for a vertical mirror.