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Kissing Triangles

Determine the total shaded area of the 'kissing triangles'.

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Isosceles

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.

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Overlap

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the blue square.

Linkage

Age 11 to 14 Challenge Level:

Four rods, two of length a and two of length b, are linked to form a kite, as shown in the diagram. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Now suppose the four rods are assembled into a linkage which makes a parallelogram. What is the maximum area of this parallelogram?