A person whose eyes are H = 1.62 m above the floor stands L = 2.21 m in front of a vertical plane mirror whose bottom edge is 40 cm above the floor, shown below. What is the horizontal distance x to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror?

Horizontal distance to floor = 0.7246 m or 72.46 cm

Explanation:

The reflection of the nearest part of the floor will be seen at the bottom part of the mirror.

Let's start by imagining there is beam starting from the person's eyes hitting the bottom part of the mirror. It will be reflected at the same angle as it arrived. To find this angle we will use the following lengths:

Vertical Distance of eyes - Vertical distance of bottom edge of mirror

= 1.62 - 0.4 = 1.22 m

This will be used with the distance from the wall (2.21 m) to find the angle of incidence of the beam.

This is:

Tan(theta) = Perpendicular/Base

Tan(theta) = 1.22 / 2.21 = 0.552036

Taking the inverse of tan to find theta we get: Theta = 28.9°

This is the angle of incidence, and the angle of the reflected ray will thus be 90° - 28.9° = 61.1°

Since the height of the mirror and angle of reflection of the beam are known, we can calculate the horizontal distance of the floor:

Tan (61.1°) = Horizontal distance to floor / height of mirror

Tan (61.1°) = Horizontal distance to floor / 0.4

Horizontal distance to floor = 0.7246 m or 72.46 cm

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