Question

A person $1.80\;m\;$ tall stands in front of a plane mirror. What is the minimum height of the mirror, and how high must its lower edge be above the floor for the person to be able to see his/her whole body? Person’s eyes are $6.0\;cm\;$ below the top of the head.

Answer 1

Understanding the answer requires an intermediate result using the law-of-reflection. When a ray of light is incident on a plane reflecting surface, the incident ray and the reflected ray can lie on the identical plane. One can define the angles of incidence and also the reflection as follows:
The angle of incidence = Angle between the incident ray and also the normal to the reflecting surface at the point of incidence.
The angle of reflection = Angle between the reflected ray and also the normal to the reflecting surface at the point of incidence.
The law of reflection states that the angle of reflection is identical to the angle of incidence.
A Useful Result: Consider a situation where a lightweight ray from an object gets reflected by a mirror and enters an eye. If the object and therefore the eye is at an identical distance from the mirror then the law of reflection implies that the point of incidence lies exactly midway between the eye and also the object.

Complete step by step answer:
Now, we will apply the above result to our problem in which a $1.8m\left( {or} \right)180cm$ tall person stands in front of a mirror. The top of her head, her eyes, and her feet, all lie along the identical vertical line. We will consider the two objects are the 'top of her head' and 'her feet'. These two objects lie at an identical distance from the mirror as her eyes.
The point of incidence of the ray from the 'top of her head' must lie at the midpoint between her eyes and therefore the top of her head. Since the eyes are $6cm$ below the top of her head, the point of incidence of the ray must be $180cm - 3cm = 177cm$ from the ground. This marks the upper fringe of the mirror.
The point of incidence of the ray from her feet must lie at the midpoint between her feet and her eyes. Since the eyes are $174cm$ from her feet, the point of incidence of this ray must be$\dfrac{{174}}{2} = 87cm$. This marks the lower fringe of the mirror.
So the lower fringe of the mirror must be $87cm$ from the ground and also the vertical dimension of the mirror must be $177cm - 87cm = 90cm$, which is half the peak of the person.

Note: The plane mirror contains a flat reflecting surface. If a parallel beam of light is incident on the plane mirror then reflected light is additionally parallel. The key use of a plane mirror is to determine reflected objects. The distance of the object from the mirror is always equal to the distance of the image formed.