Question

In the figure shown below, a ray is incident on a plane mirror. Find the value of $\theta$ for which the ray retraces its path after the third reflection.

• A. $\theta =\frac{\alpha }{2}$
• B. $\theta =2\alpha$
• C. $\theta =\alpha$
• D. $\theta =\frac{\alpha }{4}$

Answer 1

The correct option is B $\theta =2\alpha$


From the given diagram, we can deduce that, the incident ray $\text{IC}$ on plane mirror $\text{PQ}$ is parallel to the plane mirror $\text{QR}$.

For the ray to retrace along the same path, angle of incidence in the third reflection should be equal to ${90}^{\circ }$.

Let the angle between $\text{CB}$ and $\text{AB}$ be $x$.

From $△CAB$ we get, $90+90-\theta +x=180\Rightarrow x=\theta$

Hence, angle of incidence on plane mirror $\text{QR}$ is $\frac{\theta }{2}$.

$\therefore$ Angle of reflection on plane mirror $\text{QR}$ is $\frac{\theta }{2}$.

From $△QAB$, we can say that, $\alpha +90+90-\frac{\theta }{2}=180$

$\Rightarrow \theta =2\alpha$

Why this question? This question gives student a better understanding of solving problems related to ray diagram. Note:- According to triangle sum theorem, Sum of the interior angles of a triangle is always equal to ${180}^{\circ }$.