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Kissing Triangles

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


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.


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?

Constructing Triangles

Age 11 to 14
Challenge Level

Take a ten-sided die (or other random number generating tools - a pack of cards with the picture cards removed, a calculator, a phone app...) and generate three numbers. Construct a triangle using these three numbers as the side lengths.

If you're not sure how to use a ruler and compasses to construct a triangle given the lengths of its three sides, watch the video below:

Generate a few more sets of numbers and draw some more triangles.
What do you notice?

Here are some questions you might like to consider:

  • Can you draw more than one triangle from each set of three numbers?
  • When is it possible to construct a triangle from the three numbers generated?
  • Is there a quick way to tell if it will be possible to construct a triangle?

Here is a game you could play:
Start with 10 points. Roll three dice. If a triangle can be drawn, you gain a point, if it can't, you lose a point. If you reach 20 points you win the game, if you reach 0 you lose.
Which is the more likely result?

Here is a game you could play with another person:
Player A chooses an integer length between 1 and 10cm. Player B randomly generates the lengths of the other two sides. If  a triangle can be drawn, Player B wins; otherwise they lose.
Is there a "best" length that Player A should choose?
Is this a fair game?

Now explore what happens if you generate 4 numbers and draw a quadrilateral.