Final answer:
To see your entire body in a flat mirror, the mirror should be half the height of the person. The top should be at eye level, and the proportions remain the same regardless of the distance from the mirror.
Step-by-step explanation:
To see your entire body in a flat mirror, the height of the mirror should be exactly one-half the height of the person. This is because the angle of incidence is equal to the angle of reflection, and so to see your feet which are at the ground level, the top of the mirror should be at eye level. Mathematically, the lines of sight to the top and bottom of the mirror form similar triangles with your eyes and the extremities of your body. As a result, the height of the mirror is not affected by the distance away from the mirror. This can be concluded because as you move further away, although the angles change, the proportional property of the similar triangles holds true, and you will still see your entire reflection.
Therefore, if a person is 1.75 meters tall, the optimal height for the mirror would be 0.875 meters. The top of the mirror should be hung at 1.715 meters from the floor, while the bottom of the mirror should be situated 0.825 meters from the floor. These measurements ensure that the mirror is centered at the viewer's eye level for a person of this height.