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
The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?
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 activity is plain straight forward problem solving. Here
with traditional instruments. Working out what the task is asking
for, doing something which might help but isn't the whole solution,
reviewing the result of that to see what might bethere to be
noticed, and then using that observation to close the task. Along
the way working into a deeper appreciation of the things we thought
we already knew and could do easily.
Start the group off with the task of drawing a square of any
size, using only a ruler and compasses. There's plenty to challenge
understanding just in that.
Allow some discussion to establish that every student
appreciates the task and that many 'first thoughts' about the task
can be shared.
Pursue each suggestion, together as a group but with each
students actually drawing for themselves. Exploring suggestions to
their point of failure deepens understanding, skipping this reduces
the value of the activity considerably.
If a prompt towards something fruitful is needed, point out that
partially meeting criteria can sometimes be helpful in the
problem-solving process. In this case to create a square, based on
one side, one of whose 'top' corners touch one of the other two
sides of the triangle. Several of these will be needed so that we
can think about the special one of this set which touches both
Take time to let students notice how they create their squares.
How they choose the base or the 'touch point' and from that go on
to create the rest of the square. Once they have three or four
squares invite them to stand back, take a look and then share what
they notice. Ask how that helps with the actual task
This is a relatively closed task but follow the extension
suggestion to foster a classroom culture that 'plays' with
compasses and sparks with geometric reasoning and insight.
Here's a couple of nice problems to start discussion and
If I only have the three mid-points from the sides of a triangle
can I recreate the triangle ?
If I only have the centres of three circles where each circle
touches the other two can I recreate the circles.
As before, these tasks are to use ruler and compasses, but do
allow the discussion to wander into methods which depart from that
Also take time to allow students to appreciate what kind of
constraint 'ruler and compasses' imposes.
'Ruler' means tha points can be joined and the line continued
indefinitely from both ends, and 'compasses' means that a length
can be picked up and transferred somewhere else.
What, in your own words, is the challenge or task here ?
How can you use that to get the square you want ?