So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Bend

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

30-60-90 Polypuzzle

Stage: 5 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.