Similarity and congruence

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?

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

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

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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.

What is the same and what is different about these circle questions? What connections can you make?

Explore the relationships between different paper sizes.

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Can you find the distance between the two trees using the information given?

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