Question

a man stands in a room with his eyes at the centre of the room the height of the ceiling is H. the length of the shortest plane Mirror on the wall in front of the man so that the man can see the full image of the wall behind him is
1) 2H/3
2)H/2
3)H/3
4)H/4

Answer 1

The length of the mirror is one by third of the height of the wall.

(3) is correct option.

Explanation:

Given that,

Height of ceiling = H

H is the height of the wall. M is the length of the mirror.

O is the head of the observer.

According to figure,

ΔOAB & ΔOA'B' are similar.

$\dfrac{A'B'}{AB}=\dfrac{OA'}{OA}$

$\dfrac{\dfrac{L}{2}}{L+\dfrac{L}{2}}=\dfrac{1}{3}$

$\dfrac{A'B'}{AB}=\dfrac{1}{3}$.....(I)

ΔOAC & ΔOA'C' are similar.

$\dfrac{A'C'}{AC}=\dfrac{OA'}{OA}$

$\dfrac{A'C'}{AC}=\dfrac{1}{3}$.......(II)

Now from equation (I) and (II)

$\dfrac{A'B'}{AB}=\dfrac{A'C'}{AC}=\dfrac{1}{3}$

$\dfrac{A'C'+A'B'}{AC+AB}=\dfrac{1}{3}$

Substitute the value

$\dfrac{M}{H}=\dfrac{1}{3}$

$M=\dfrac{H}{3}$

Hence, The length of the mirror is one by third of the height of the wall.

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Topic : Optics

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