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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?

Napkin

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem
 

This problem offers a simple context which can generate lots of questions. Inviting learners to make conjectures and form convincing arguments. Demonstrating the similarity of triangles is relatively straight forward and calculating lengths offers opportunities for links with Pythagoras' theorem and ratios, bringing together important geometrical concepts.

Possible approach
 

Time to engage in, and become familiar with, the context is important. Early on, encourage learners to list and share what they notice, using large squares of paper on a display board can encourage discussion of key features and ideas and conjectures which they might explore.

Identify questions about the triangles that learners will work on.

As learners work this document may help them discuss possibilities and focus on some possible approaches.

It is likely that learners will arrive at results in different ways. These journeys and findings form opportunities to share and discuss good and elegant solutions and different ways of "seeing".
 

Key questions

  • What do you think might be true?
  • What do you know?
  • What do you need to know?
  • What mathematical ideas and techniques might be of use in order to answer that question?
     

Possible extension

If the square paper napkin is folded so that the corner P does not coincide with the midpoint of an opposite edge, where would you place the fold for a 5, 12, 13 or an 8, 15, 17  or a 7, 24, 25 triangle?

Are any of these findings extendable to other quadrilaterals?


See also the problem Origami
 

Possible support
 

Making Sixty is a similar context involving a similar fold but with more accessible results. It also has the potential to lead to some practical mathematics.