Question

A person dressed in a new suit stands in front of
a plane mirror fixed on a vertical wall. Height of the
person is 'H' and that of his eyes from the ground
is 'h'. Show that minimum length of the mirror so
that the person can have his full view is H/2 and
that it is independent of the position of eyes. Also
determine the position of mirror relative to ground
and show image formation with a ray diagram.

​

Answer 1

Plane Mirror should be 1/3 of the height of wall, so that to see the complete wall of height H in the mirror.

Explanation:

Let HB be the wall with height h and MM' is the mirror.

A man standing at distance x from mirror and the wall is at distance y from man.

Let MM' be r,  

Then,

EA=EG=HF=DB = MM' = r                                  ---(1)

In Δ RMB, C is midpoint of BA, and CA is parallel to RM

A is midpoint of BM.

I is the midpoint of QH, G is midpoint of HM'

In Δ MFB, A is midpoint of BM, and AE is parallel to BF.

E is midpoint of MF.                     (midpoint theorem)  

In Δ M’DH, E is midpoint of M'D

Therefore,

2EA = FB and 2GE = HD

⇒ HD = FB = 2r                                                         ---(2)

From equations 1 & 2, we get

HD = HF+HD

r + FD = 2r

FD = r

∴ HD = FD = BD = r

HB=HD + FD + BD = 3r

h = 3r

r = $\frac{h}{3}$

Hence, to see the complete wall infront of height H in the mirror, the person will require plane mirror of 1/3 of height of wall.