Similarity and congruence

Explore the relationships between different paper sizes.

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

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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?

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .