Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

Pythagoras for a Tetrahedron

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

Calculating with Cosines

Stage: 5 Challenge Level:

Try to make sense of each method one line at a time. The following mathematical topics may be useful:

Pythagoras' theorem

Cosines of angles greater than $90^{\circ}$

Distance between two points on a coordinate grid

Trigonometry in right-angled triangles