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Use time to discuss what different groups notice and how it might help. Encourage rigour.
This problem creates a good opportunity to practise some common classroom geometry constructions, for example: a perpendicular bisector of a given line, angle bisector centre for the circumcircle of a given triangle, drawing a line through a given point parallel to a given line.
It is particularly important to promote a lively discussion of possible reasons for a construction method's validity.
Establish the relationships with paper folding and cutting and use this to see why triangles are congruent and therefore why particular approaches are valid.
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?