# Math Help - Mirrors

1. ## Mirrors

Hey guys,

I got a question here:

Facing a slightly steamed-over mirror, hold one eye shut and trace the outline of your face in the mirror. Explain why the outline is exactly 1/2 the width and 1/2 the height of your face.

I have no idea how to approach this question and anyyy help would be greatly appreciated.

2. Hello GreenDay14
Originally Posted by GreenDay14
Hey guys,

I got a question here:

Facing a slightly steamed-over mirror, hold one eye shut and trace the outline of your face in the mirror. Explain why the outline is exactly 1/2 the width and 1/2 the height of your face.

I have no idea how to approach this question and anyyy help would be greatly appreciated.
Study the attached diagram.

The line $AB$ represents the width of your face, with $E$ as your right eye. $A'B'$ is the reflection of your face in the mirror, which lies along the line $MPQ$.

Then it is well known that the distances of the object and its reflection from the mirror are equal. So
$AM = MA'$

$\Rightarrow AM = \tfrac12AA'$
A ray of light from $A$ hits the mirror at $P$ and is reflected into your eye, $E$, producing an image of $A$ at $A'$. So $EPA'$ is a straight line, with $P$ the apparent point on the mirror of the position of $A'$.

Similarly for $Q$: this is the apparent point on the mirror of the position of $B'$.

By similar triangles, since $AM = \tfrac12AA'$, we can easily prove that $PQ = \tfrac12A'B' = \tfrac12AB$.