Problem Teachers' Notes Hint Solution

# Cyclic Quadrilaterals

This problem is in two parts. The first part consists of four similar challenges which provide building blocks to help you to solve the final challenge. You could work on them with others, but if you are working on your own, you may not need to attempt all four.
Of course, you are welcome to go straight to the Final Challenge!
In this problem, you will be working with cyclic quadrilaterals. A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. There is an interactivity at the bottom of the page which you can use to create cyclic quadrilaterals.
Click on one of the questions below to get started.
Question A
Using a circle with nine evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these nine-point circles useful.
How many different triangles can you find?
Work out all the angles in these triangles.
On a nine-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples:
Create at least six different cyclic quadrilaterals on nine-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals.
What do you notice about the angles on opposite vertices of your quadrilaterals?
Question B
Using a circle with twelve evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these twelve-point circles useful.
How many different triangles can you find?
Work out all the angles in these triangles.
On a twelve-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples:
Create at least six different cyclic quadrilaterals on twelve-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals.
What do you notice about the angles on opposite vertices of your quadrilaterals?
Question C
Using a circle with fifteen evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these fifteen-point circles useful.
How many different triangles can you find?
Work out all the angles in these triangles.
On a fifteen-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples:
Create at least six different cyclic quadrilaterals on fifteen-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals.
What do you notice about the angles on opposite vertices of your quadrilaterals?
Question D
Using a circle with eighteen evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these eighteen-point circles useful.
How many different triangles can you find?
Work out all the angles in these triangles.
On an eighteen-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples:
Create at least six different cyclic quadrilaterals on eighteen-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals.
What do you notice about the angles on opposite vertices of your quadrilaterals?
Final Challenge
What can you say about the angles on opposite vertices of any cyclic quadrilateral?
Use a diagram like the one below to create a convincing argument.
Does your argument still work if the centre of the circle is not within the quadrilateral?
You may find this Virtual Geoboard useful.
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