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Kissing Triangles

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

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Hallway Borders

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

Dividing a Square

Age 11 to 14 Short Challenge Level:
Three lines have been drawn from the centre of this square to split it into two congruent trapezia and a pentagon.

The side length of the square is 1, and the areas of the three shapes are all equal.

What is the length $x$?

This problem is adapted from the World Mathematics Championships
You can find more short problems, arranged by curriculum topic, in our short problems collection.