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?
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.
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.
problem requires the solver to reason geometrically and make
use of symmetry. By re-presenting the information in a different
way, for example by adding additional lines (a useful technique in
geometrical problems) more structure can be revealed. It is an
interesting idea that adding something, and therefore apparently
making it more complex, can sometimes make a problem more
accessible. Then of course there is an opportunity to use the
cosine rule in a non-standard context.
Use the images in this
document to make cards.
Display the problem and ask learners to work in groups
rearranging the cards in ways which help to make connections for
Share ideas and relationships that groups notice before going on
to solve the problem. Encourage careful reasoning and convincing
arguments. For example:
There are several ways of solving the problem so share different
approaches and discuss what helped to move thinking forward.
Two observations may be worth drawing attention to :
Ask the students if they can explain why these should all be