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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

### Long Short

A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot be greater than sqrt2. Find a quadrilateral of this type for which s2 is approximately sqrt3and show that s2 is always less than sqrt3. Find a quadrilateral of this type for which s3 is approximately 2 and show that s2 is always less than 2. Find a quadrilateral of this type for which s4=2 and show that s4 cannot be greater than 2.

### Diagonals for Area

Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.

### Why do this problem?

First, the surprising amount of variation in possibilities shown in the video is worth the journey. Secondly, even though there is such variation the outer quadrilateral is always cyclic.Finally, specialising by trying numbers can help form a map of the journey you need to make in order to prove the generalisation for any cyclic quadrilaterals.

### Possible approach

The first stage is simply to investigate:
Construct an image with the given constraints either using dynamic geometry or with a ruler and compasses. Using ruler and compasses is difficult simply because you need some flexibility to ensure a reasonable overlap of the circles.

Are learners surprised by the flexibility visible in the dynamic image?
Allow time for lots of discussion about construction techniques, the order of working (formed by the constraints) and the freedoms available (how many circles will meet the cirteria?).

Now for the problem.
A first step is to encourage exploration by writing in some angle sizes (following a discussion of the properties of opposite angles of a cyclic quadrilateral). Does the outside quadrilateral have opposite angles whose sum is 180 degrees and is therefore cyclic?

In specialising by using numbers for angles and keeping track of which angles can be calculated from others, the steps to a generalisation are much clearer.

### Key questions

• What defines a cyclic quadrilateral?
• What are the freedoms?
• What the contraints?