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Kissing Triangles

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

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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?

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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?

Shapely Pairs

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Tom worked really hard on these problems. He used the triangle cards. Here's what he sent us.
How many possible pairs? I worked this out by picking a card, working out how many pairs could be made with it using the remaining cards, and then discarding the card.

The answer is 46, obtained with the following breakdown:

Has all its angles equal: 3 (Does not contain a right angle, Has all its sides equal, Has three lines of symmetry)
Does not contain a right angle: 9 (Has all its sides equal, All its angles are of different sizes, Has three lines of symmetry, Has just 2 equal sides, Has only 1 line of symmetry, All its sides are of different lengths, Does not contain a right angle and has just 2 equal sides, Has no line of symmetry, Has just 2 equal angles)
Does not contain a right angle and has just 2 equal sides: 3 (Has just 2 equal angles, Has just 2 equal sides, Has only 1 line of symmetry)
Has three lines of symmetry: 1 (Has all its sides equal)
Has all its sides equal: 0
Contains a right angle: 10 (Has just 2 equal angles, Has just 2 equal sides, All its sides are of different lengths, Has no line of symmetry, All its angles are of different sizes, Contains a right angle and has just 2 equal angles, Contains a right angle and has just 2 equal sides, Contains a right angle but does not have a line of symmetry, Contains a right angle and has all its sides of different lengths, Has only 1 line of symmetry)
Contains a right angle and has just 2 equal sides: 4 (Has only 1 line of symmetry, Has just 2 equal sides, Has just 2 equal angles, Contains a right angle and has just 2 equal angles)
Has just 2 equal angles: 3 (Has only 1 line of symmetry, Contains a right angle and has just 2 equal angles, Has just 2 equal sides)
Has just 2 equal sides: 2 (Has only 1 line of symmetry, Contains a right angle and has just 2 equal angles)
Contains a right angle and has just 2 equal angles: 1 (Has only 1 line of symmetry)
Has only 1 line of symmetry: 0
Has no line of symmetry: 4 (Contains a right angle but does not have a line of symmetry, Contains a right angle and has all its sides of different lengths, All its angles are of different sizes, All its sides are of different lengths)
Contains a right angle but does not have a line of symmetry: 3 (Contains a right angle and has all its sides of different lengths, All its angles are of different sizes, All its sides are of different lengths)
All its sides are of different lengths: 2 (All its angles are of different sizes, Contains a right angle and has all its sides of different lengths)
All its angles are of different sizes: 1 (Contains a right angle and has all its sides of different lengths)
Contains a right angle and has all its sides of different lengths: 0


It is possible to be left with 4 cards at the end, for example:
Has all its angles equal
Does not contain a right angle and has just 2 equal sides
Contains a right angle and has just 2 equal sides
Contains a right angle but does not have a line of symmetry
(and the rest can indeed all be paired).


It is also possible to pair all of the cards. For example:
Has three lines of symmetry, Has all its sides equal
Has all its angles equal, Does not contain a right angle
Contains a right angle but does not have a line of symmetry, Has no line of symmetry
Has just 2 equal angles, Has just 2 equal sides
Contains a right angle and has just 2 equal angles, Contains a right angle and has just 2 equal sides
Contains a right angle and has all its sides of different lengths, All its sides are of different lengths
Contains a right angle, All its angles are of different sizes
Has only 1 line of symmetry, Does not contain a right angle and has just 2 equal sides.


It is possible to have 6 cards such that any 2 form a pair. For example:
Contains a right angle
Contains a right angle and has just 2 equal sides
Contains a right angle and has just 2 equal angles
Has only 1 line of symmetry
Has just 2 equal angles
Has just 2 equal sides