Compared to a person's height, what is the minimum length (top to bottom) of a mirror that will allow the person to see a complete image from head to toe?

Draw a diagram representing a person standing in front of a mirror vertically mounted on a wall. Label their height as H and the mirror's length as L_m. Place the person at a distance D from the mirror. Draw rays of light traveling from their head and feet to the mirror and back to their eyes.

Label the point where the light ray from the person's head hits the mirror as A, the reflection point on the mirror for the feet's light ray as B, and the person's eye level as E. Since the angle of incidence equals the angle of reflection, the triangle formed by points A, E, and the intersection of the light rays traveling from the head to the mirror and from the mirror to the eye, is similar to the triangle formed by points B, E, and the intersection of the light rays traveling from the feet to the mirror and from the mirror to the eye. Name these intersection points C and D, respectively.

Based on the similar triangles, we can set up a proportion as follows: \(\frac{A H_{1}}{D} = \frac{B H_{2}}{D}\) where \(AH_{1}\) is the distance of the light ray's path from the head to the mirror and \(BH_{2}\) is the distance of the light ray's path from the feet to the mirror.

By simplifying the proportion from step 3, we obtain: \(\frac{A H_{1}}{B H_{2}} = 1\) As we are looking for the mirror's length, we know that \(L_{m} = AH_{1} - BH_{2}\). We can multiply both sides of our equation by \(AH_{1} - BH_{2}\): \(\frac{A H_{1}(AH_{1} - BH_{2})}{B H_{2}} = L_{m}\)

As the triangles are similar, we can establish the following relationship: \(\frac{AH_{1}}{H} = \frac{BH_{2}}{H - E}\) We are interested in finding the mirror's length. Therefore, we have to eliminate \(AH_{1}\) and \(BH_{2}\) from our equations. Then, we can find the length \(L_{m}\) in terms of the height \(H\) and the eye level \(E\).

Solving the equation from step 5 for \(AH_{1}\) and \(BH_{2}\), we get: \(AH_{1} = \frac{H}{H - E}BH_{2}\) Now we can substitute this expression back into the equation from step 4: \(\frac{\frac{H}{H - E}BH_{2}(AH_{1} - BH_{2})}{B H_{2}} = L_{m}\) After some simplification, we obtain: \(L_m = \frac{H(\frac{H}{H - E}-(E - B))}{2}\) Since the person's eyes are exactly at the center of their height, E should be equal to \(\frac{H}{2}\). Substituting this into our final formula, we get: \(L_m = \frac{H(\frac{H}{H - \frac{H}{2}}-(\frac{H}{2} - B))}{2}\) Now we see that independent of the person's distance to the mirror (which is represented by B), the minimum length of the mirror can be half of the person's height, \(L_m = \frac{H}{2}\). Therefore, the minimum length of a mirror that allows a person to see their full reflection is half of their height.