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Six Discs

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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Trapezium Four

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

Why do this problem?

This problem has been created to draw attention to area relationships within any trapezium.

Possible approach:

Encourage students to play with the form and extract (propose and support with reasoning) as many properties as possible.
Students may need to draw a variety of trapezia until they have a mental map of the situation. This will give them confidence and help them see the generality within the problem.

Key questions:

  • Have you made a number of different trapezia and found the area of each of the four triangles? (Dynamic geometry software may be useful here)

  • What did you find?

  • What are the cases to be considered in the problem? [all four areas different, two matching, three matching, and all four equal] How would the trapezium have to be to make each case occur?

Possible extension:

What is the ratio of the four triangle areas? start with some specific lengths for the parallel sides and the distance between them if that helps.

Possible support:

Students who are not ready for this challenge without preliminary activity might explore a figure made from two parallel lines of length 3cm and 5cm, at a perpendicular distance of 4cm from each other. Lines are drawn between the left end of the 3cm line and the right end of the 5cm, and similarly with the other ends, to create two similar triangles. Students can then explore the ratio between lengths in the figure.
  • What can you say about these two triangles?
  • What can you say about their areas?
  • What can you say about line ratios in this figure?
  • How much freedom to move does this figure possess, if the conditions involving the lengths 3cm, 4cm and 5cm are maintained? And how are the answers above altered by that movement?