Question

The cross-section of a prism has the form of an isosceles triangle. One of the equal faces is silvered. A ray is normally incident on the other onc's face and being reflected twice inside the prism emerges through the base of prism perpendicular to base. Angle of prism is found to be $\pi / n$. The value of $n$ is________

Answer 1

Let angle of prism is $\alpha$ and other two angles are $\beta$ each. As $N N^{\prime}$ is parallel to $P Q$.
$\angle P Q R=\angle Q R N^{\prime}$ (Complimentary angles)
$\Rightarrow 2 \alpha=\beta$

Now from $\triangle A B C \alpha+2 \beta=180^{\circ}$
$\alpha+2(2 \alpha)=180^{\circ}$
$5 \alpha=180^{\circ}$
$\alpha=36^{\circ}=\frac{\pi}{5}$