1

Label carefully each of the following:

sectorcircumference

arcradius diameter chordmajor segment minor segment

centre

labelling activity

tangent

CircleGeometry

Board of Studies

2

CircleGeometry

Definitions of circle, centre, radius, diameter, arc, sector, segment, chord, tangent, concyclic points, cyclic quadrilateral, an angle subtended by an arc or chord at the centre and at the circumference, and of an arc subtended by an angle should be given.

These are the terms the Board of Studies says you should know for this topic:

question:Do we know them all?

You should also know what a secant is!

3

CircleGeometry

Equal arcs on circles of equal radii subtend equal angles at the centre, and conversely.

(Conversely means that the opposite is also true, i.e. that equal angles at the centre subtend equal arcs on circles of equal radii!)

While you do not HAVE to be able to prove this....you should be able to do so.

question:How?

Assumption:

4

CircleGeometry

Equal angles at the centre stand on equal chords and converse

Theorem: proof not examined

Almost the same as the last (assumption), chords drawn from A to B and D to J are equal.

question:Can you construct

the proof?

5

CircleGeometry

Theorem: proof not examined

The angle at the centre is twice the angle at the circumference subtended by the same arc.

geogebra activity

x

x y

y

Proof:

6

CircleGeometry

Two circles touch if they have a common tangent at the point of contact.

Theorem: proof not examined

7

CircleGeometry

Theorem: proof not examined

The tangent to a circle is perpendicular to the radius drawn to the point of contact, and converse.

question:

Why?

geogebra activity

8

CircleGeometry

Theorem: proof IS examinableThe perpendicular from the centre of a circle to a chord bisects the chord

Proof:AO=BO (radii circle)AC = common side

9

CircleGeometry

Theorem: proof IS examinable

The line from the centre of a circle to the midpointof a chord is perpendicular to a chord

This follows logically from the previous theorem

Proof:OX=OY (radii of circle)OY=common sideXY=ZY (Y midpoint of XZ)So by SSS rule, ∆OXY ≡ ∆ OZY

10

CircleGeometry

Theorem: proof IS examinable

Equal chords in equal circles are equidistant from the centres,and converse.

geogebra activity

Proof: Let CD=AB

11

CircleGeometry

Any 3 non-collinear points are concyclic . They lie on a unique circle, with centre at the point of intersection of the perpendicular bisectors of the intervals joining these points.

Theorem: proof IS examinable

To see this concept clearly, experiment using the 'circle through three points' option in geogebra.

A

B

C

A

B

C

O

Proof:The perpendicular line through the midpoint

of AB goes through the centre of the circle (see theorem slide 7). Similarly with BC and AC. Where the three perpendicular bisectors meet MUST be the centre of the circle.

Pull

Pull

12

CircleGeometry

Theorem: proof IS examinableAngles in the same segment are equal

Proof:Join A and D to centre O

13

CircleGeometry

Theorem: proof IS examinable

The angle in a semicircle is a right angle

geogebra activity

Proof: Follows on from angle at centre = 2 x angle at circumference. If angle at centre = 1800, angle at circumference must be 900.

Pull

Pull

14

CircleGeometry

Theorem: proof IS examinable

Opposite angles in a cyclic quadrilateral are supplementary

geogebra activityProof:Join B and D to OObtuse

15

CircleGeometry

Theorem: proof IS examinable

The exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite angle

geogebra activity

Proof: Let

16

CircleGeometry

Theorem: proof IS examinable

If an interval subtends equal angles at two points on the same side of it then the endpoints of the interval and the two points are concyclic

geogebra activity

17

CircleGeometry

Axiom: A tangent to a circle is perpendicular to the radius at that point. Converse is also true.

geogebra activity

18

CircleGeometry

Theorem: proof IS examinable

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Firstly: What does 'alternate segment' mean?

geogebra activity

DF is a tangent to the circle at C. CE is a chord dividing the circle into two segments, CEA and CEB.

The segment CEA is said to be alternate to

19

CircleGeometry

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Proof:Draw in diameter CF and join EFLet

20

CircleGeometry

Theorem: proof IS examinable

Tangents to a circle from an external point are equal

geogebra activityProof:OA=OB (radii)OC = common side

21

CircleGeometry

Theorem: proof IS examinable

The products of the intercepts of two intersecting chords are equal

geogebra activity

Proof:

22

CircleGeometry

Theorem: proof IS examinable

The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point

PQ2 = QR QS

Proof:

23

CircleGeometry

Theorem: proof IS examinable

When circles touch, the line between the centres passes through the point of contact'

geogebra activity

Proof:AB is a tangent to circle centre O

24

Groves

25

Pull

Pull

26

27

Pull

Pull

28

29

Pull

Pull

30

31

Pull

Pull

32

Groves

33

Pull

Pull

34

35

36

Pull

Pull

37

Groves

38

39

Pull

Pull

40

Groves

41

42

Pull

Pull

43

44

45

46

Pull

Pull

47

Fitzpatrick

48

49

50

51

Pull

Pull

52

53

CircleGeometryEnd of topic:

• Summary including theorems and examples. Make sure you can use them AND construct the proofs where appropriate.• You don't need to complete every question in this file. Make sure you have done a number from the last exercise though. You will probably need me to check them.• Make some post-its of theorems and stick them on mirrors and walls!

Attachments

jw_Angles_standing_on_arc.ggb

jw_line_from _centre_to_midpoint_chord_perpendicular_to_chord.ggb

jw_perp_bisector_chord_passes_through_centre.ggb

17a_CONSTRUCT_Tangent_Perpendic_to_Radius.ggb

5a_CONSTRUCT_Common_Chords.ggb

1_Chords_From_Centre.ggb

jw_Angle_in_semicircle.ggb

jw_Opp_angles_cyclic_quadrilateral.ggb

10_Ext_angle_cyclic_quad.ggb

alternate segment circle theorem.ggb

tangent meets radius.ggb

11_Tangents_from_external_point.ggb

14_Product_of_intercepts_of_chords.ggb

16_Tangent_squared_equals_secant_intercepts_products.ggb

jw_Centres_of_touching_circles.ggb

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra.xml

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geogebra.xml

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geogebra.xml

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geogebra.xml

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geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra.xml

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geogebra.xml

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geogebra_thumbnail.png

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

Page 1: definitionsPage 2: Jan 1-3:15 PMPage 3: = arcs = angles at centrePage 4: < at centre = < on = chordsPage 5: < at centre= 2 x < circPage 6: touching circlesPage 7: tangent perp. to radiusPage 8: perp bisects chordPage 9: perp to chord bisects chordPage 10: = chords equidistant centrePage 11: 3 concyclic pointsPage 12: