Question

A man is $180\text{cm}$ tall and his eyes are $10\text{cm}$ below the top of his head. In order to see this entire height right from toe to head, he uses a plane mirror kept at a distance of $1\text{m}$ from him. The minimum length of the plane mirror is

• A. $180\text{cm}$
• B. $90\text{cm}$
• C. $85\text{cm}$
• D. $170\text{cm}$

Answer 1

The correct option is B $90\text{cm}$

Total height of man $\left(H\right)=180\text{cm}$
(Including eyes)
Distance between head and eyes,
$HE=10\text{cm}$
Distance between feet and eyes,
$FE=170\text{cm}$


Considering man as an extended object in front of mirror, man will be able to see his entire image when the light incident on mirror from his head and feet reaches his eyes after reflection.

$\Rightarrow$ From the symmetry of incident and reflected rays$(\angle i=\angle r)$
$HM=EM\&EP=PF$

At points $M\&P$ of mirror, $MO$ and $PD$ will be the perpendicular bisectors of length $HE$ and $EF$ respectively.
$\Rightarrow NP=ED=\frac{EF}{2}$
and, $MN=OE=\frac{EH}{2}$
Minimum required length of mirror will be,
$L_{\text{min}}=NP+MN$
$\Rightarrow L_{\text{min}}=\frac{EF}{2}+\frac{EH}{2}$
$\therefore L_{\text{min}}=\frac{170}{2}+\frac{10}{2}=90\text{cm}$
Hence, option $\left(b\right)$ is the correct answer.

Alternate:
Minimum length of the plane mirror $=\frac{\text{Height of the man}}{2}=\frac{180}{2}=90\text{cm}$

Key Point: Minimum length of the plane mirror is half the height of the man standing in front of it.