Multiple Reflections With Inclined Mirrors

Geometry Level 2

The angle between two mirrors is θ \theta . If the incident ray that is parallel to mirror 1 is reflected 5 times in the setup, and becomes parallel to mirror 1 again. What is the measure (in degrees) of θ \theta ?


The answer is 30.

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Iit Jee Aspirants by Iit Jee Aspirants , 34, India

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5 solutions

Ajit Athle
Jun 6, 2014

Total Deviation = 360 - 5(θ) which in this case =180 + θ or 6θ=180° or θ=30°

6 Helpful 1 Interesting 0 Brilliant 1 Confused

Total deviation is 180+ theta how can it identify

Madhuri Dholakiya - 7 years ago

@Madhuri: I've drawn a diagram which traces the exact path of the ray and explains the solution much better but I do not know how to upload the diagram. If you would write to me at ajitathle@gmail.com, I can forward it to you.

Ajit Athle - 7 years ago

For Incident and Reflection angles to be equal, the condition can only be met for five reflection if there is a symmetry. And that means after third reflation the ray would trance back the same path. That means second reflection on m-2 must be at 90 degrees. If you draw the ray diagram you would see that the other two angles are Alpha/2 and Alpha = 90. Hence Alpha/2 = 30 .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

thus.. there will be no two different parallel rays... the given pic is not realistic.

Ananya Aaniya - 4 years, 9 months ago
Sudipan Mallick
Jun 7, 2014

thita=pi/n+1 (where n is no. of refections) here n=5 so thita = 180/5+1=30.

3 Helpful 0 Interesting 1 Brilliant 0 Confused

Please correct the typo. π n + 1 = 180 5 + 1 = 30. \dfrac\pi {n+1}=\dfrac{180}{5+1}=30.

Niranjan Khanderia - 4 years, 9 months ago

What is this formula about can u elaborate ?

Tarun B - 3 years, 8 months ago
Isaac Browne
Aug 20, 2017

Here is a really nice way of thinking of the reflections.

From here it is obvious that θ = 180 / 6 = 3 0 ∘ \theta = 180/6=30^{\circ}

1 Helpful 1 Interesting 0 Brilliant 0 Confused

An explanation I found elsewhere:

https://socratic.org/questions/two-plane-mirrors-are-included-at-an-angle-theta-as-shown-in-the-figure-light-ra

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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