# Mirror, mirror, on the wall – to see all of me, you should be how tall?

15 Aug 2016399 Views

Once upon a time, in a kingdom far, far away, there lived a wicked queen…

You already know this story. The queen was obsessed with her appearance. The fable tells us that the queen was driven to murderous jealousy when her mirror told her that it was her stepdaughter Snow White, and not she, who was the fairest of them all.

Whatever the reason behind her excessive vanity, there’s no denying that it was a force to be reckoned with. The queen’s mirror was, therefore, her most-prized possession.

But the pair wasn’t always inseparable. There was a time in the queen’s life when the magical mirror didn’t exist at all. In fact, it had to be custom-built for her.

The queen needed to be sure that she could see every inch of her frame, from the top of her head (B) to the tips of her toes (A).

Can you use maths to find, in terms of the queen’s height (AB), the minimum length that the craftsperson needs to make the mirror so that the queen can see her whole body?

Scroll down for the solution to this week’s puzzle.

This week’s maths puzzle comes courtesy of Dr Bernd Kreussler, lecturer in mathematics at Mary Immaculate College in Limerick, who is actively involved in mathematics enrichment classes at the University of Limerick and the Irish Mathematical Olympiad.

Main image via Shutterstock

Image via Shutterstock

## Solution:

The mirror needs to be at least half as tall as the wicked queen.

When the queen visited the craftsperson who was to make her famous mirror, she insisted that she see all calculations. This had to be perfect.

The craftsperson drew this diagram for the queen.

Explaining it to the queen, the craftsperson said, “You will see the tips of your toes at point P, and the top of your head at point Q.

“We know that light hitting a perfectly flat surface reflects from it at an angle equal to that at which it hit the surface. We know that my mirrors are flawless, and therefore perfectly flat.

“Therefore, we know that the triangle formed by your line-of-sight and the imaginary line from the top of your head to where you see it reflected is isosceles. The same goes for the triangle formed between you and where you see the tips of your toes.

“Knowing that these are isosceles, we can see that Q – the top of your head – is opposite the midpoint of BE, and P is opposite the midpoint of AE.

“Therefore, PQ is equal to AB divided by 2. So the mirror has to be half your height.”

And so it was.

Kirsty Tobin was careers editor at Silicon Republic

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