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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.

Ratty

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Trice

Age 11 to 14
Challenge Level

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.

To find the area of the triangle PQR, Bithian Heung's solution use Pythagoras' theorem twice and shows that triangle PQR is an equilateral triangle.

$$BQ = \sqrt {BG^2+ GQ^2} = \sqrt {3^2+ 1^2} = \sqrt {10}$$
$$PQ = \sqrt {PB^2 + BQ^2} = \sqrt {2^2 + 10} = \sqrt {14}$$

Similarly $QR = RP = \sqrt {14}$ and hence PQR is an equilateral triangle.

It is useful to remember that an equilateral triangle splits into two 30-60-90 degrees triangles. By Pythagoras theorem again the ratio of the sides is 1: $\sqrt 3$ : 2.

Hence the area of the triangle PQR is $\frac {1}{2} (7\sqrt 3)$ square units.