Why do this problem?

This problem provides suitable challenges for different age groups. It demonstrates a powerful inter-relationship between geometry and algebra. The interactivity enables learners to experiment and make conjectures.

If you start with the simple case of a triangle and then later consider the pentagon, it is accessible for younger learners when it can be used to practise using the midpoint formula and solving simultaneous equations.

For older students it provides an exercise in linear algebra and the study of conditions for the existence of solutions tosystems of equations.

The problem also links to the problem Polycircles which has a different context but where the mathematics is identical giving an experience of isomorphism in mathematics.

Possible approaches

For younger learners who know how to solve a pair of simultaneous linear equations this problem provides a good series of challenges. Taking a numerical example where the coordinates of the midpoints of the sides of the triangle are known (see the Hint), and the coordinates of the vertices have to be found, three simultaneous linear equations can easily be found and solved. Because the coefficients are all unity the equations are easy to solve. Even though learners may only have been taught to solve two simultaneous equations in two unknowns many will be able to solve these three equations for themselves and get satisfaction from being able to do so independently. The results can be checked by drawing.

The next step is to generalise from a particular numerical example and to use exactly the same steps in the algebra to derive formulae for the vertices of atriangle, and then a pentagon, in terms of the midpoints of the edges.

The interactivity will suggest that the problem for quadrilaterals does not generally have solutions. The next challenge is to explain why this is so.

The problem came from an Oxford entrance paper dated 1926 which did not mention coordinates but asked candidates to construct the pentagon given only the midpoints. It might be interesting to discuss in class whether students today would use the same methods as students in 1926.

Possible support

For a geometrial method start with Triangle Midpoints

For an algebraic method, the hint gives a numerical special case to support younger and less confident learners.

Possible extension

You could pose the problem for general polygons and leave the learners to decide for themselves whether or not to start with special cases. The problem Polycircles is a natural extension of this problem.