Question

Corridor figure shows an overhead view with a plane mirror Mounted at one end. A burglar B sneaks along the corridor directly toward the centre of the mirror. If d = 6cm, how far (in m) from the mirror will she be when the security guard S can first see her in the mirror?

Answer 1

For a plane mirror, the distance between the image and the reflecting surface of the mirror is equal to the distance between the object and the mirror. Use this and with help of geometry find the distance of the image.

Complete answer:
Let us first understand how an image of an object is formed with a plane mirror.
When an object is placed in front of a plane mirror, the light emitted by the object falls on the mirror and the light rays are reflected according to the laws of reflection.
The light rays reflect in such a way that all the rays appear to be emerging from a point behind the mirror. This point is the position of the image.
For a plane mirror, the distance between the image and the reflecting surface of the mirror is equal to the distance between the object and the mirror.
Suppose the distance between mirror and the burglar is x when the security guard sees her for the first time in the mirror.
Then the image of the burglar and the reflecting surface of the mirror is x.
Let us find the value of x.
The security can see the image when the rays from the image enter his eyes. Follow the figure below.

Consider the triangles $\Delta IMO$ and $\Delta SDO$.
Here, $\angle IOM=\angle SOD=\theta$ and $\angle IMO=\angle SDO={{90}^{\circ }}$.
Hence, $\Delta IMO\approx \Delta SDO$ with AA similar triangles rule.
Since $\Delta IMO\approx \Delta SDO$, we get that $\dfrac{IM}{SD}=\dfrac{MO}{DO}$.
$\Rightarrow \dfrac{x}{d}=\dfrac{\dfrac{d}{2}}{d}$
$\Rightarrow x=\dfrac{d}{2}=\dfrac{6}{2}=3cm$
Hence, the burglar is at a distance of 3cm from the mirror when the security guard S can first see her in the mirror.

Note:
Instead of finding the distance of the image from the reflecting surface of the mirror, we can directly find the distance of the burglar from the mirror with the flowing ray diagram.

Here also the triangles are similar and in the similar way we will get the value of x = 3cm.