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30-60-90 Polypuzzle

Age 16 to 18 Challenge Level:

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Can you re-arrange the pieces of the puzzle to form a rectangle by sliding the pieces without rotating them? Now can you re-arrange the pieces to form an equilateral triangle by flipping the pieces numbered $2$ and $5$ and moving them into new positions?

You can assume that pieces $1$ and $5$ each have a side of length one unit, that the pieces as shown form a perfect square of area one square unit and that they do fit together to form a perfect equilateral triangle of the same area. This will tell you that some of the angles are $60^{\circ}$, some are $30^{\circ}$ and some are $90^{\circ}$.

Calculate the length of the edge of piece $3$ which is labelled '$t$' and then calculate the lengths of all the other edges giving answers correct to $3$ significant figures.