Pythagoras' theorem

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?

Describe how to construct three circles which have areas in the ratio 1:2:3.

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.