AN INTRODUCTION TO CIRCLE THEOREMS – PART 2

Slideshow 47, Mathematics

Mr Richard Sasaki, Room 307

OBJECTIVES

• Review circle properties• Learn some properties

regarding angles and circles

THE CIRCLE

Let’s learn and recall some basic circle property names.

Centre (origin)

Radius

Tangent

THE CIRCLEDiameter

Chord

Sector

Radii (plural of radius)

Centre (origin)Central

angleArc length

CIRCLE PROPERTIESSo far we know…

A tangent is always 90o to its radius.

2a

a

An angle at the edge is half the angle at the centre.

b

a

For a cyclic quadrilateral, opposite angles add up to 180o.

PROPERTY 4For a triangle with the diameter of the circle as an edge, the opposite angle touching the circle’s edge is a right-angle.

You should have showed this before on the worksheet!

180o We can see this as a quadrilateral with an 180o

angle.

PROPERTY 5In circles, angles in the same segment are equal to one another.

2a

a

We know the central angle is twice the angle at the edge.a

The position at the edge makes no difference.So the angles at the edges are equal.

PROPERTY 5In circles, angles in the same segment are equal to one another.

aa

Be careful, nothing here is congruent! They are similar though. a

a

ANSWERS

PROPERTY 6The last we’ll learn. An angle between the tangent and a chord is equal to the angle in the alternate segment.

𝑦

𝑥

First, label two we know are right-angles.Label .

90−𝑥 Internal angles in a triangle: 𝑦+90+90−𝑥=180

𝑦+180−𝑥=180𝑦−𝑥=0𝑦=𝑥

PROPERTY 6Actually, for this property to work, the chord doesn’t need to pass through the origin.

First add two radii. One that touches the tangent, the other that touches another vertex.

𝑥

𝑦2 𝑦90− 𝑦

90−𝑦

The triangle is isosceles. If one angle is , the other two are…180−2 𝑦

2=90−𝑦

Lastly on a line, we get .Simplifying this, we get .

PROPERTY 6An angle between the tangent and a chord is equal to the angle in the alternate segment.

𝑥

𝑥

ANSWERS

b.

c.

2.

3.