# Circumscribed Triangles Investigation

- Author:
- Ian Radcliffe

This investigation will be assessed against Criteria B and C. Remember that you will need to write and Introduction and Conclusion. These can be written last. You will also need evidence for your answers. This includes screen clips and data.
Introduction:
Spend some time playing with the points on the circle to see what happens. Make sure you understand what is in the algebra column and how it is connected to the diagram.
Part A: Circle Theorems [Criterion B Levels 1-4]
1.
(a)Compare the size of Angle A and the Central Angle BOC.
• Move point B around and record different values for both angles. Record your data in a table.
• What do you notice? Explain using sketches and your data.
(b) Repeat this for the following pairs of angles:
• Angle B and Central Angle COA
• Angle C and Central Angle AOB
Remember to record your data and explain any patterns
(c) Write conjectures summarising any patterns you found.
2.
(a) Move point C so that the line “a” goes straight through the center of the circle.
• What do you notice about the Angle A?
(b) Move the point C back, and then move point A so that the line “c” goes through the center of the circle.
What do you notice about Angle C?
(c) Make line “b” go through the center and comment on the size of Angle B.
(d) Write conjectures describing any patterns you found.

Part B: Sine Rule and circle radius[Criterion B Levels 5-6]
1. You are going to compare the sine rule with the radius of the circle. First you will look at comparing the ratio a/sin A and the circle radius.
(a) Changing “a”
• Draw a table with the following headings : “a”, “sin(AngleA)”, "a/sin(AngleA)" and “radius”
• Change the size of “a” and record the measurements. You will need to explain how many measurements you decided to take!
(b) Repeat, but this time change the radius
(c) Repeat , but this time change the Angle A.
(d) Explain any patterns you find.
2. You now need to check your patterns to see if they work for the ratio "b/sin(AngleB)” and the "radius".
• Did you find the same result? ( use sketches and data to explain)
• Write a formula that summarises your findings.
3. Check that your formula works by using the last angle ratio and the radius: "c/sin(AngleC) and "radius".
Part C: Proof [Criterion B Levels 7-8]
You now need to prove that your formula is correct for ALL circumscribed triangles:
• Use one of the triangles from Part A Question 2 and a trigonometric rule.
Hints:
(a) A "proof" should be done without any numbers
(b) Label the sides of the triangles with the "a", "b" and "c", instead of numbers.
(c) Label the angles "A", "B" and "C", instead of numbers.
(d) Think about how long the hypotenuse is in terms of the radius of the circle (Part A Question 2).
(e) Remember you want your rule to look the same as the one in Part B.