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How Big?

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Hex

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

Overlap

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the blue square.

Triangular Tantaliser

Age 11 to 14 Challenge Level:

Here is a 4 by 4 dotty grid:
Picture of a Geoboard with four rows of four pegs

How many different triangles can you draw by joining dots on the grid?
We count two triangles as the same if you can cut one out and fit it exactly on top of the other.

Can you come up with a convincing argument that you have found all the possible triangles?