Ray problem (geometry)

Question

Problem:

Two plane mirrors $OP$ and $OQ$ are inclined at an acute angle (diagram is not to scale). A ray of light $XY$ parallel to $QO$ strikes mirror $OP$ at $Y$. The ray is reflected and hits mirror $OQ$, is reflected again and hits mirror $OP$ and is reflected for a third time and strikes mirror Magic Square Association $OQ$ at right angles at R, as shown. The distance $OR$ is $5$cm. The ray XY is d cm from the mirror $OQ$. What is the value of $d?$

My thoughts: the answer says its 5, but I am not sure how to specifically construct the triangles or angles and sides to be able to solve the question. I did write a bunch of angles but Im not sure how to proceed. Can anyone help me solve this problem?

Answer 1

Let the first, second and third reflection take place at the points $Y, S$ and $T$. respectively. Define point $V$ to be the foot of the perpendicular from $Y$ to $OQ$.

Observe, $$\theta=\angle TOR=\angle OYS=22.5^{\circ}~~\text{and} ~~\angle SYV=\angle YSV=\angle TSR=45^{\circ}$$ Hence, $$TR=RS=5\tan \theta\implies \tan\theta=\frac{VY}{OV}=\frac{d}{5+5\tan\theta+d}$$

Answer 2

Let $\angle PYX= \theta$. Then $\angle AYC=90^\circ-2\theta \implies \angle YAC=2 \theta \implies 4\theta=90^\circ, \theta=22.5^\circ$

Next, $d=YA \sin 2\theta, YA=AB \cot \theta, AB=\frac{BR}{\sin 2\theta}, BR=5 \tan \theta \implies d=5$

Answer 3

In problems involving mirrors it often becomes much easier if you draw the reflected world behind the mirror, because then you can just ignore the mirror as the rays go straight through it into the mirror world.

In the picture below, Q is reflected in P to give Q', P is reflected in Q' to give P', and Q' is reflected in P' to give Q''. The light ray travels straight through these mirror worlds and meets Q'' at right angles. By the way, this shows that the angle between the mirrors must have been 90/4=22.5 degrees (though in the original drawing the angle is a bit off so the reflections in the picture below are a little wonky). It is now also obvious that OR=OR'=OR''=d.

Answer 4

The total angle reflected ( $\gamma= 90 ^{\circ} $) is given. we could unfold a stack to obtain a "development" of the reflection scenario as given:

Let the number of mirrors be $m$. The number of points of incidence is also $m$. Draw a sector on a thin transparency sheet dividing total angle $\gamma$ by $m+1$ into so many sectors or vertical angles.

Denting each reflecting edge by a sharp steel edge or needle and alternatively folding along the green reflecting mirror edges makes a stack given in the question which can be unfolded to get back the "development."

Even a print on paper can be folded from this rough sketch as in Origami.