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Triangles in a Square
Age 11 to 14
Challenge Level 
One method for working out the areas of the triangles is the 'boxing in' method:
Box in the chosen shape, work out the area of the box, and then subtract the unwanted areas in order to work out the desired area.
The example below shows how this method can be used to work out that the area of the tilted square is 10 cm$^2$.
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Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
Linkage
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?