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'Pentagon' printed from http://nrich.maths.org/
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
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
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
which has a different context but where the
mathematics is identical giving an experience of isomorphism in
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.
For a geometrial method start with
For an algebraic method, the hint gives a numerical special case to
support younger and less confident learners.
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
is a natural extension of this problem.