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

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

$\displaystyle \Rightarrow AM = \tfrac12AA'$

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

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

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

Grandad