Circles Theorems

1
Circles Theorems

• Revision

2
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.

3
Circle
Tangent
Circumference
Radius
Diameter
4
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?
5
Angles on the Circumference

• http//www.mathsnet.net/dynamic/car/circle03.html

6
Angles subtended by the same chord are equal
Chord
7
Angles subtended by the same chord are equal
The chord is not always drawn in.
8
Angles subtended by the same chord are equal
What is the value of a? a 35?
35?
a
9
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?
10
Angle at the Centre

• http//www.mathsnet.net/dynamic/car/circle04.html

11
Angle subtended by the same chord at the centre
is twice that at the circumference
a
2a
12
Angle subtended by the same chord at the centre
is twice that at the circumference
Sometimes you will have to spot the property
being used.
a
2a
13
Angle subtended by the same chord at the centre
is twice that at the circumference
What is the value of angle a?
25?
2a 180? 50? 2a 130? a 65?
50?
a
a
14
Rayner Higher Page257 Ex21
15
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?

16
Cyclic Quadrilaterals

• http//www.mathsnet.net/dynamic/car/circle01.html

17
The opposite angles of a cyclic quadrilateral add
up to 180?
a
A cyclic quadrilateral is a quadrilateral whose
vertices all lie on the circumference of a circle.
b
Angle a angle b 180?
18
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?
19
Angles in a Semi-circle

• http//www.mathsnet.net/dynamic/car/circle02.html

20
Angle in a semi circle equal 90?
21
Rayner Higher Page 259 Ex 22
22
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?
23
The radius and tangent are at right angles
o
24
The radius and tangent are at right angles
o
25
The radius and tangent are at right angles
o
26
The radius and tangent are at right angles
o
27
The radius and tangent are at right angles
o
28
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?
29
Tangents to a Circle

• http//www.mathsnet.net/dynamic/car/circle06.html

30
Two tangents from the same point to a circle are
equal
31
Rayner Higher Page 261 Ex 23Mixed
QuestionsRayner Higher Page 262 Ex 24