# Reflecting in mirror

## Homework Statement

A. Draw accurately in the figure the light beam that goes from Object A to eye B after being reflected on the mirror. It must be consistent with the mirror principle!

B. At question A. you may have connected the point behind the mirror (A') with eye B.
And found the correct light path that way.

b. Will that also work if you use the point behind the mirror of B (B') instead? Do you get the same result?
Use mathematical arguments for your judgement!

## Homework Equations

mirror reflection rules

## The Attempt at a Solution

Question A:
I tried to draw point A' and connected that to B. The purple line is the normal line (the imaginary line that is perpendicular to the mirror) And the angle between the mirror and A is the same as the angle between the mirror and B.

Question B: I think this is true, I can draw the same lines, but I don't know what is meant by 'mathematical arguments'

## Answers and Replies

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cepheid
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When they say use mathematical arguments, they mean to use what you know about geometry. Call the point where the ray strikes the mirror point M. For part a, you can show that since the angle between line segment MA and the mirror is the same as the angle between line segment MA' and the mirror, it must be true from basic geometry that the angle between segment MB and the mirror is the same as these other two angles.

What if you drew the line segment going to B' instead? How would this argument go?

-So do you need to proof similarity or congruence?

I guess You can show that the angle between line segment MB and the mirror is the same as the angle between line segment MB' and the mirror.

"it must be true from basic geometry that the angle between segment MB and the mirror is the same as these other two angles." Don't you mean those 3 other angles? Since there are 4 angles in total and you only mentioned the one between MB and the mirror.

When they say use mathematical arguments, they mean to use what you know about geometry. Call the point where the ray strikes the mirror point M. For part a, you can show that since the angle between line segment MA and the mirror is the same as the angle between line segment MA' and the mirror, it must be true from basic geometry that the angle between segment MB and the mirror is the same as these other two angles.

What if you drew the line segment going to B' instead? How would this argument go?

cepheid
Staff Emeritus
Science Advisor
Gold Member
-So do you need to proof similarity or congruence?

I guess You can show that the angle between line segment MB and the mirror is the same as the angle between line segment MB' and the mirror.

"it must be true from basic geometry that the angle between segment MB and the mirror is the same as these other two angles." Don't you mean those 3 other angles? Since there are 4 angles in total and you only mentioned the one between MB and the mirror.
Huh? I explicitly mentioned three angles in my post:

1. Angle between segment MA and mirror
2. Angle between segment MA' and mirror
3. Angle between segment MB and mirror

What I was saying was that for part A, you can use geometry to show that the third one in the list is equal to the first two (which were "those other two angles").

How many similarity proofs do you need? Do you to proof that all 4 triangles are similar? (A-mirror-M is similar to A'-mirror-M is similar to B-mirror-M is similar to B'-mirror-M) Or can you do it in less steps?

cepheid
Staff Emeritus
Science Advisor
Gold Member
Well you can prove that triangle A-mirror-M is similar to triangle A'-mirror-M, and that's how you know that the angle between segment MA and the mirror equals the angle between segment MA' and the mirror.

However, I don't think you need a similarity proof for the last bit, because you have this property:
http://en.wikipedia.org/wiki/Vertical_angles

I don't even know whether your teacher wants a rigorous geometric proof or not. It might be sufficient to just state that these angles are equal (EDIT because this is a known result of geometry)

The real point I was trying to make in my first post was just that if you draw the point at B' instead of A', you can still use the exact same sequence of arguments to show that angle of incidence = angle of reflection.