Why do this problem?
is all about geometrical reasoning and
proof. It gives learners the opportunity to play with a
problem and come up with a conjecture from practical
demonstrations, and then justify their findings. It also opens up
discussion of when it is necessary to write something as a strict
This problem lends itself to investigation with a dynamic
geometry tool such as Geogebra.
The learners could start by constructing a quadrilateral on a
unit circle, and displaying the lengths of each side. The challenge
is to make the shortest side as long as possible. Once the length
has been found, learners need to come up with a convincing argument
that it is not possible for the shortest side to be any
Learners could then discuss in pairs the best way of making the
second side as large as possible. What happens to the shortest side
in this process? The question says "Side $b$ must be
less than a certain value" so there is the opportunity to discuss
why it has to be strictly less than that value and can never
actually reach it.
The last part of the question encourages discussion along similar
lines. It is good to bring out of the discussion that the third
side can be as close to $2$ units as you wish but can never
actually be exactly $2$.
If I move the points on the circumference to increase one of
the sides, what will happen to the adjacent side?
What shape should I make in order to make the shortest side as
long as possible?
The same problem could be tackled as a hexagon on the circumference
of a circle, rather than a quadrilateral.
Start by considering a triangle on the circumference of a circle
and calculate the maximum and minimum side lengths.