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 trees 20 metres and 30 metres long, lean across a passageway between two vertical walls. They cross at a point 8 metres above the ground. What is the distance between the foot of the trees?

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

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

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?