Question

A boy is standing in front of an erect plane mirror. His uncle is standing behind him as shown. The height of the boy is $3\text{feet}$ and that of the uncle is $6\text{feet}$. What is the minimum length of the mirror required so that the boy can completely see his uncle’s image in the mirror?

Answer 1

Let AB represent the uncle, CD represent the boy and GF represent the mirror.
The boy can see complete image of his uncle in the mirror when rays of light from point A and B, after reflection from mirror (GF), reach the point C (eye of the boy) as shown in the figure.
Let H be any point on the mirror such that GH $=x$ and HF $=y$.
Now, AB $=6\text{feet}$, CD $=3\text{feet}$

Al $=3-x$
FE $=3-y$
CH $=3\text{feet}$
BE $=$ IG $=6\text{feet}$
In ΔHCG and ΔAGI,
$\frac{GH}{CH}=\frac{AI}{IG}$
$\frac{x}{3}=\frac{3-x}{6}$
$x=1feet$

$In\Delta CFHand\Delta BFE$,
$\frac{HF}{CH}=\frac{EF}{BE}\frac{y}{3}=\frac{3-y}{6}y=1feet$
$=$ GF $=x+y=(1+1)\text{feet}=2\text{feet}$

The minimum length of mirror required by the boy to see his uncle’s image is $2\text{feet}$.