question_answer

A ray of light is incident at \[50{}^\circ \] on the middle, of one of the two mirrors arranged at an angle of \[60{}^\circ \] between them. The ray then touches the second mirror, get reflected back to the first mirror, making an angle of incidence of

A) \[50{}^\circ \]                          

B) \[60{}^\circ \]

C) \[70{}^\circ \]                          

D) \[80{}^\circ \]

Correct Answer: C

Solution :

Let required angle be \[\theta \] From geometry of figure In \[\Delta ABC, \alpha  =180{}^\circ  - \left( 60{}^\circ  + 40{}^\circ  \right) = 80{}^\circ \] \[\Rightarrow \,\,\,\,\,\beta \,\,=\,\,90{}^\circ  -80{}^\circ  =10{}^\circ \] \[\operatorname{In}\,\,\,\,\,\,\,\,\,\,\Delta \,ABD, \angle A = 60{}^\circ , \angle B=(\alpha +2\,\beta )\] \[= \left( 80 + 2 \times  10 \right)\,\,=\,\,100{}^\circ \,\,and\,\,\angle D- (90{}^\circ  -\theta )\] \[\because \,\,\,\,\,\,\,\angle A+\angle B+\angle D=180{}^\circ \] \[\Rightarrow \,\,\,\,\,\,60{}^\circ  +100{}^\circ  +\left( 90{}^\circ  -\,\,\theta  \right) =180{}^\circ \] \[\Rightarrow \,\,\,\,\,\,\,\,\theta =70{}^\circ \]