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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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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?

Rati-o

Age 11 to 14 Challenge Level:
An additional challenge here is to show that the inscribed polygon is a parallelogram. Invite students to make conjectures about the shape of this quadrilateral and then to prove their conjectures.