A child and a man are standing in front of a plane mirror as shown in the figure. What should be the minimum height of the mirror so that the child can completely see his/her own image as well as the man's image in the mirror.

The correct option is D $\frac{5H}{6}$
For the child to see its full image, the length of mirror required is $\frac{H}{2}$. But, to see the full size image of the man, the child requires an extra length "$x$" of the mirror as shown in the figure. Let the total length of the mirror be $\ell .$
Also from the geometry it is clear that no extra mirror is required for the child to see the image of man's foot.



From $△RQM:$

$\tan \theta =\frac{RQ}{QM}=\frac{H-x}{2d}......\left(1\right)$

From $△M{M}^{\prime }{P}^{\prime }:$

$\tan \theta =\frac{M{M}^{\prime }}{M^{\prime }{P}^{\prime }}=\frac{x}{d}.......\left(2\right)$

Using $\left(1\right)$ and $\left(2\right)$ we get,

$\frac{H-x}{2d}=\frac{x}{d}\Rightarrow x=\frac{H}{3}$

Total length of mirror, $\ell =x+\frac{H}{2}$

$\Rightarrow \ell =\frac{H}{3}+\frac{H}{2}=\frac{5H}{6}$