Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Bendy Quad

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.

Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

Xtra

Stage: 4 and 5 Challenge Level:

Why do this problem?

There are several different ways of solving this problem. It is a good example to bring home the point that, to be an expert problem solver, and to understand a piece of mathematics, it is not enough to find an answer. One should ask oneself "Have I used the best method?". This problem can be solved using trig formulae and the cosine rule but it can be solved more straightforwardly using only Pythagoras theorem.

Possible approach

Challenge the class to find different methods of solution.

Key question

How do we interpret the two solutions?

Is the diagram correct for both solutions?

Isosceles triangles. Quadratic equations. Pythagoras' theorem. Complex numbers. Continued fractions. Golden ratio. Mathematical reasoning & proof. Circles. Surds. Cosine rule.

You may also like