Question

A man is standing exactly midway between a wall and a mirror and he wants to see the full height of the wall (behind him), in a plane mirror (in front of him). If the height of the wall is H, then the minimum length of the mirror should be
$\begin{align}
  & a)\dfrac{H}{4} \\
 & b)\dfrac{2H}{3} \\
 & c)\dfrac{H}{3} \\
 & d)\dfrac{H}{5} \\
\end{align}$

Answer 1

Hint: It is to be noted that a person can see the full image of an object if and only if the rays hitting the extreme points of the object reach the human eye. Hence we will draw the ray diagram in such a manner that the man stands in between the wall of height H and the mirror. Then from the ray diagram accordingly we will obtain the required relation between the height of the mirror and the height of the wall.

Complete step by step answer:

In the above figure we can see that the man is standing between the wall AB and the mirror FE. The rays of light from the extreme of the wall reach the man to his eye at point G. Since the mirror is a plane mirror the object and the image are formed at the same distance from the mirror i.e. d. We can also imply that the height of the object and the height of the image is the same i.e. H.
If we consider triangle DGC and triangle FGE, both the triangles are similar by AA similarity criteria. Since two triangles are similar their ratio of the corresponding sides are equal. Hence we can write,
$\dfrac{FE}{DC}=\dfrac{GI}{GK}$
But the height of the wall i.e.\[CD=AB=H\text{ },\text{ }GI=d/2\text{ }and\text{ }GK=3d/2\] . Let us say the height of the mirror FE=M. Therefore we can write the above equation as,
$\begin{align}
  & \dfrac{M}{H}=\dfrac{GI}{GI+IK} \\
 & \Rightarrow \dfrac{M}{H}=\dfrac{d/2}{3d/2} \\
 & \Rightarrow \dfrac{M}{H}=\dfrac{1}{3} \\
 & M=\dfrac{H}{3} \\
\end{align}$

Hence the correct answer of the above question is option c.

Note:
It is to be noted that if one wants to see the complete image of anything then we must consider the rays from edges reaching our eyes by reflection from the mirror. This is because if we are in a position of viewing the top and the bottom of a building, then we can say that we can view the entire image of the building. It is to be noted that for a plane mirror to view the image does not depend on the distance between the mirror and the wall.