Similarity and congruence

Can you sort these triangles into three different families and explain how you did it?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

Can you make sense of these three proofs of Pythagoras' Theorem?

Why does this fold create an angle of sixty degrees?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Two ladders are propped up against facing walls. At what height do the ladders cross?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?