circles are all around us. they are products of both our natural environment and of our artificial...

Circles are all around us. They are products of both our natural environment and of our artificial environment.

Circles are part of the astronomically big

Whirlpool Galaxy

Model of an Atom

Throughout history circles have been considered to have a certain perfection and harmony to them. Even today there is a certain mystic and wonder around the circle.

and the infinitesimally small.

CIRCLE TERMINOLOGY

To study circles, it is important that we are familiar with the terminology of circles.

O

A circle is defined by a group of points all being the same distance from a point identified as the centre, O.

A circle defines three distinct groups of points:

- the points inside the circle (including centre).

- the points on the circle.

- the points outside the circle.

O

Radius – a line segment drawn from the centre to any point on the circle.

RadiusA

C

B

Chord – a line segment between any two points on a circle

Note: A line segment is a line which has a beginning and an end (the endpoints) and consequently it has a defined length and can be measured.

When naming line segments, the two endpoints are always used in either order.

OA

BC

O

Diameter – a chord that passes through the centre of a circle.

D

E

DE

Arc – a series of points on the circumference between two given points on the circumference.

Arcs are classified into two groups: Minor arcs – arcs that make up less than one half of the circumference.Major arcs – arcs that make up one half or more of the circumference.

3 letters must be used to name a major arc (two letters to identify the ends of the arc and a third letter between them to identify any other point on the arc).

E

D

F

B

A

M

DFE EFDor not

DE

AMB

or AB

When naming arcs,Minors arcs can be named using 2 letters representing the endpoints (although the 3 letter designation can also be used as with major arcs).

Line – a series of points along a straight line that go infinitely in both directions.To name a line you can use any two points on the line

L

MN

Secant – a straight line that intersects a circle at two points.

AB

LN NLor MNorTangent – a straight line that intersects a circle at one point. BAor BTor ATor

Point of Tangency –

point of intersection which is shared by the tangent line and the circle. T

A

B TT

A

B

DISTINCTION BETWEEN CHORDS AND ARCS

A

B

A

B

Chords

Just as Criminals Steal Apples so do

However it does NOT make sense to say that:

In the same way it doesn’t make sense to say that:

Arcs Subtend Chords

Apples Steal Criminals

AB

AB

subtends

therefore

AB

AB

but

does not subtend

Subtend Arcs

A

B

When referring to chords, radii, arcs, diameters, secants and tangents, it is important to use the correct symbol above the letter names.- used for line segments (chords, radii or

diameters)- used for lines (tangents or secants)

- used for arcs (minor or major)

A

AB AB

The 3 geometric items above are all referring to different sets of points.

BAB

AB

AB

AB

- refers to all points on the circumference between A and B- refers to all points on a straight line between A and B- refers to all points on a straight line between and beyond A and B

Angles are also elements of a circle that are often referred to. An angle is a rotational separation between two rays. The angles of a circle are classified into 4 categories based on the location of their vertices:

(1) Central angle – vertex at the centre

(2) Interior angle – vertex inside the circle

(3) Exterior angle – vertex outside the circle

(4) Inscribed angle – vertex on the circumference

BO

A

AOB is a central angle because its vertex, O, is at the centre of the circle. Incidentally, it is also an interior angle because its vertex is inside the circle. However, just as we don’t refer to a square as a rectangle (eventhough it is) so do we not refer to a central angle as an interior angle.

AOB intersectsAB

-formed by two radii

O

DEF is an interior angle because its vertex, E, is inside of the circle.

F

D

E

G

H

DEF intersects and its vertical angle, GEH intersects

DFGH

-formed by two chords

OM

I

KL

J

N

IKM, IKN and MKN are all exterior angles because their vertex, K, is outside of the circle.

IKM intersects and IM JL

-formed by two secants, two tangents or a secant and a tangent

MNMKN intersects andLN

INIKN intersects and JN

O

QPT, QPR and RPT are all inscribed angles because their vertex, P, is on the circumference of the circle.

R

QP

T

S

QPT intersects

QPR intersects

RPT intersects

QRP

QR

RSP

-formed by two secants or a secant and a tangent

КОНЕЦfinal τέ�λο

σfinitoThe end

le fin

sofپايا

ن