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Fitting In

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

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Look Before You Leap

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?

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Two Ladders

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?

Sitting Pretty

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem :

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.

Possible approach :

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.

  • What possible values can x and y have and does the relationship hold in these cases?
  • Do they notice anything about the relationship between x and y?
  • How many possible values of x and y can there be, and does the relationship hold for any of them?

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.

Key questions :

  • Look at your diagram, what's there that might help ?
  • What's there anyway ?
  • Paying attention to any triangles in your diagram, what do you think about the relationship between them. What might be true ?
  • Is it ? Why ?
  • How might that help ?
  • What is it that this problem asks us to find out ?

Possible extension :

The Line and Its Strange Pair

Possible support :

Some students might not immediately be ready for this because they are not sufficiently familiar with the similar triangle context, but it is more likely that the algebra demand is too great. It may be that this problem works well as a 'guided example' where the teacher still asks questions, and shares the 'blocks', which at each point need to be identified and a way ahead sought, but takes the group on a pre-determined journey to the solution. Perhaps asking learners to work with some real values to explore possibilities and verify the relationship for these cases.