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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.

Trice

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Tetrahedra Tester

Age 11 to 14
Challenge Level

What are the limitations on the lengths of the sides of any triangle?

Look at different cases.

If the base is a 4 by 5 by 6 triangle, what size can the other faces be?

How many different tetrahedra can have as its base a 4 by 5 by 6 triangle?

Can you find 8 or more possible tetrahedra?

How will you know when you have them all?