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?
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.
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
If I tell you two sides of a right-angled triangle, you can easily work out the third using Pythagoras' theorem. But what if the angle between the two sides is not a right angle? Is there a way to work out the length of the third side?
Some students tried to work out a formula to work out the length of the third side when we know two sides and the angle between them.
Below, you can see the start of their methods. Can you finish each method to produce a formula?
Does each method lead to an equivalent formula?
Does each method work for both acute- and obtuse-angled triangles?