Similarity and congruence

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

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

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

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.