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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.

### Lawnmower

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

### Long Short

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

### Why do this problem?

This problem provides an opportunity for rich discussion of properties of quadrilaterals and circles, and leads to geometrical reasoning in searching for proofs and counter-examples.

### Possible approach

This printable worksheet may be useful: Circles in Quadrilaterals.

Show the three examples of tangential quadrilaterals and allow the learners to identify what they have in common. Share the definition of a tangential quadrilateral as one where a circle can be constructed inside to just touch all four sides.

Allow plenty of time to experiment - this could be through sketches on mini whiteboards or rough paper at first, leading to more accurate construction using ruler and compasses or using dynamic geometry. Encourage pairs to share their conjectures about in which quadrilaterals it will be always, sometimes or never possible to inscribe a circle.

Bring the class together to share their thoughts. Expect some disagreement, particularly for those shapes which are only sometimes possible. Give learners time to work in small groups to produce posters showing some or all of the quadrilaterals with reasoned arguments to explain their conclusions about whether they are tangential quadrilaterals or not, and then ask groups to present their conclusions to each other, encouraging healthy debate where disagreement still exists.

### Key questions

When is it possible to draw a circle inside a kite? a trapezium?
Are there any quadrilaterals where it is never possible to inscribe a circle?

### Possible support

Create lots of diagrams to build up ideas of what is and isn't possible. There is a diagram in the Hint showing a semicircle constructed in a triangle; considering this may help for those quadrilaterals which can be cut along a line of symmetry into two triangles.

### Possible extension

If the side lengths of a tangential quadrilateral are $a$, $b$, $c$ and $d$, with $a$ opposite $c$ and $b$ opposite $d$, show that $a+b = c+d$.

Bicentric Quadrilaterals builds on the ideas discussed in this problem and links with work on cyclic quadrilaterals.