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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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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?

Shapely Pairs

Age 11 to 14 Challenge Level:
You're going to have to come up with a system to make sure that you don't miss out any pairs, and to make sure that you don't count any twice. You could pick one card and see how many other cards could be paired with it, to get you started.

Is it possible to pair all the cards? Try playing around with different possibilities. You might have to pair up some cards first, if they don't have many possible partners.