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
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
A visualisation can be grasped quite easily when someone points
it out, but it is more satisfying and much better for the students'
development if they gradually feel their way around the structure
with moments of revelation.
Invite learners to create tilted squares of their own, identify
coordinates of diagonally opposite corners. Can they usethese to
help to find areas? Share ideas and generalisations as they
Connections may take time to emerge and different insights might
result in different approaches. For example the area of the tilted
square might be found through considering one of the squares and a
rectangle or seen as half way between the areas of the smaller and
larger squares. Give space for learners to find their own
visualisation and share different ideas and approaches.
One important configuration to watch for is this one: