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Kissing Triangles

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

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.

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?

Disappearing Square

Age 11 to 14
Challenge Level

Several teachers didn't like this trick but is it really a trick? When the rotation takes place the grid lines up along the slopping sides but does it exactly fit? When you count the squares the second time what are you counting? The counting may not be accurate. Are the 'half' squares all exactly half?

Many of you were puzzled by this. What is really going on here? We can't 'make' an extra square of area just by chopping a shape up and putting the pieces together differently so what is happening? William from Tattingstone describes it like this:

William's solution.

I think he is trying to tell us that the rectangle C doesn't quite fit in there. The line on the sloping side of the triangle isn't really a straight line is it? Measuring is never absolutely accurate is it?