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Euclidean Geometry Circle Investigations

Euclidean Geometry Circle Investigations

1 Investigations

Investigation 1

1) Draw a circle on a sheet of paper.

2) Draw a inside the circle.

3) Find the of the chord.

4) Connect the midpoint of the chord with the of the circle. Your figure should look something like this:

5) Now measure the between the chord and the coming from the centre.

6) Compare your results with the other pupils in your group/class.

7) Try to formulate a theorem from the information you have obtained and write it down.

8) Making a clever construction see if you can prove that your theorem will be true in every case. (Hint: make use of congruency.)

9) Now for a thought experiment: Consider what will happen if you draw a line perpendicularly from the centre onto a chord. In other words if your figure looks something like this:

2 Investigation 2

1) Copy the following circles carefully:

a a a O O O O a b b b b

2) Before you measure any of the , first write down what each of these four figures above have in common.

3) Compare the circles in (1) with these two circles and then write down in which ways the circles in (1) are different to those in (3).

a a

O O. b b

4) Now measure the angles labelled a and b in (1) and (3).

5) Discuss your results with the others in your group/class and see if you can formulate a theorem.

6) Consider the first circle in (1). By making a construction through the angle at the and the angle at the centre (connecting the vertices of the two) see if you can prove that your theorem will be true in every case.

3 Investigation 3

1) Consider the following sketch and see if you can predict what the size of angle a will be. You must be able to explain your reasoning. Construct the sketch (if necessary) and test your prediction.

a

O

2) Using the information you obtained, when would you be able to prove that any given line is a ?

4 Investigation 4

1) Copy the following circles carefully:

a c a

c b b d d

2) Before you measure any of the angles, first write down what each of these two figures above have in common.

3) In which way are they different to all the other figures you have been studying until now?

4) Compare the circles in (1) with the circle below and then write down in which ways the circles in (1) are different to this one.

c a

b d

5) Now measure the labelled angles in (1) and (4).

6) Discuss your results with the others in your group/class and formulate a theorem.

7) Consider whether you can use the information you have obtained to prove that a figure such as this one, lies on a circle (in other words that all four vertices lie on the circumference of a circle). What conditions must this figure fulfil before you can draw a circle around it?

5 Investigation 5

The word is one which you have met before. It comes from the Latin word quattuor meaning ‘four’. Therefore a quadrilateral is a four-sided figure. The word cyclic comes from the Greek word kyklos meaning ‘circle’. That means that a (or cyclic quad for short) is a four-sided figure whose vertices all lie on a circle.

1) Draw at least one circle with a cyclic quadrilateral. Measure each of the angles at the vertices.

2) Compare your angles with the others in your group/class, discuss them and try to find some pattern.

3) Formulate the resultant theorem.

4) Using one of the theorems you have already proven, prove that this one will be true in all cases. You will have to make a construction.

5) This is a very useful theorem to use in proving a quadrilateral is cyclic. What requirement will a quadrilateral have to fulfil in order to be cyclic? 6 Investigation 6

1) Copy the following circle carefully: a

b

2) Before you measure any of the angles, first try to work out what the relationship between a and b will be, giving reasons.

3) Now measure the angles a and b.

4) With the help of the others in your group/class formulate the theorem.

7 Investigation 7

1) Draw a circle with a . Connect the of contact (where the tangent touches the circle) with the centre of the circle. Now measure the angle between the tangent and the .

2) Do exactly the same again, except this time make it just a chord and not a radius. Measure the angles between the chord and the tangent.

3) What is your conclusion?

4) Formulate the theorem.

8 Investigation 8

1) Copy the following circle carefully:

2) Measure each tangent from the point of contact with the circle to where the two meet.

3) Write down your conclusions and formulate the theorem.

9 Investigation 9

1) Draw a circle with a whose three vertices are on the circumference of the circle. Draw a tangent at the point of contact of one of the vertices, so that it looks like this:

2) Measure all the angles you can find in the circle.

3) Write down the values you have obtained and discuss it with the others in your group/class. Try to find patterns/similarities.

4) Formulate the resultant theorem.