The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF.
Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ
Can you spot a cunning way to work out the missing length?
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second wall.
At what height do the ladders cross?
This problem offers students a valuable context in which to visualise the effect of constraints (the fact that the centre of the circle is on the hypotenuse). They can be encouraged to establish the relationships within the context (for example by utilising properties of similar triangles), and then find a way to
use those to make a route to a solution, which might include working backwards as well as algebraic manipulation.
This printable worksheet may be useful: Sitting Pretty
Invite the group to visualise the circle, perhaps rolling into place. Draw attention to its size and position and identify the constraints under which this circle exists.
You might wish to invite learners to try fixing the radius of the circle (at 3cm say) and constructing triangles around it.
Ask students to create their own diagram. The questions below at the right moment may help to steer the thought process with a light touch. Adding lines that represent the constraints of the situation will be helpful, so students must feel free to redraw their diagrams until they are happy that they have arrived at a good representation.