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This problem has been created to draw attention to area relationships within any trapezium.
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.
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?
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 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.
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.