Family Of Circle

By : Murtadha Rafaai (0916063)

Nur Hakimi Bin Mat @ Ahmad ( 0914009)

Aims

• To show the following on a diagram of a circle: radius, diameter, sector, segment, chord and tangent.

• To introduce the circle theorems.• To find unknown angles within a

circle and at tangents to circles using the circle theorems.

Circle

Circumference

RadiusDiamete

r

Tangent

Angles subtended by the same chord

Draw in a chord

Then draw two triangles, using the chord as a base, that touch the circumference.

Measure the angles at the circumference. What do you notice?

Angles subtended by the same chord are equal

Chord

Angles subtended by the same chord are equal

The chord is not always drawn in.

Angles subtended by the same chord are equal

What is the value of a?

a = 35○

35○

a

Angle at the Centre

Draw a triangle with all the vertices touching the circumference.

Draw another triangle using the same base but with a vertex at the centre.Measure the angle at the circumference.Measure the angle at the centre.What do you notice?

Angle subtended by the same chord at the centre is

twice that at the circumference

2a

a

Angle subtended by the same chord at the centre is

twice that at the circumference

2a

a

Sometimes you will have to spot the property being used.

Angle subtended by the same chord at the centre is

twice that at the circumference

25○

What is the value of angle a?

a a50○ 2a = 180○ –

50○

2a = 130○

a = 65○

Cyclic Quadrilaterals

Cyclic Quadrilaterals• Draw a quadrilateral

with all four vertices touching the circumference.

• Measure opposite angles and find their sum.

What do you notice?

The opposite angles of a cyclic quadrilateral add up

to 180○

A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle.

a

b Angle a + angle b = 180○

Angles in a Semi-circle

Draw in a diameter.

Use this as a base to draw a triangle with the vertex at the circumference.

Measure the angle at the circumference.

What do you notice?

Angle in a semi circle equal 90○

Radius and Tangent

Draw in a radius

Draw a tangent meeting the circle at the end of the radius.

Measure the angle between the radius and the tangent.

What do you notice?

The radius and tangent are at right angles

o

The radius and tangent are at right angles

o

The radius and tangent are at right angles

o

The radius and tangent are at right angles

o

The radius and tangent are at right angles

o

Tangents to a Circle

Mark a point outside of the circle.

Draw two tangents to the circle from this point

Measure the lengths of the tangent from the point to the circle.

What do you notice?

Two tangents from the same point to a circle are

equal