Chord Properties

1
Chapter 7 The Circle
7.2
Chord Properties
7.2.1
MATHPOWERTM 11, WESTERN EDITION
2
The Circle - Definitions
A circle is the set of all points in the plane
that are equidistant from a fixed point, the
centre. A chord is a line segment that joins two
points on the circle. A diameter is a chord that
passes through the centre of the circle. A secant
is a line that contains a chord.
7.2.2
3
The Circle - Definitions contd
7.2.3
4
The Perpendicular Bisector of a Chord
Theorem The perpendicular bisector of a chord
contains the centre of the circle.
The line segment joining the centre of the circle
and the midpoint of a chord is perpendicular to
the chord.
The perpendicular bisectors of two non-parallel
chords intersect at the centre of the circle.
7.2.4
5
The Perpendicular Bisector of a Chord
The perpendicular bisector of a chord contains
the centre of the circle.
Prove that AC BC.
AO BO
Radii
O
Reflexive
OC OC
Right Angles
B
C
A
HS
Thus, AC BC.
Congruent Triangles
7.2.5
6
The Perpendicular Bisector of a Chord
Find the required lengths to the nearest tenth.
1. Find the radius
Given The length of the chord AB is 12 units and
a line passing through the centre of the circle
is a perpendicular bisector of AB.
r
A
5
12
B
c2 a2 b2 52 62 61 c 7.8
Therefore, the radius of the circle is 7.8 units.
7.2.6
7
The Perpendicular Bisector of a Chord
2. Find the length of the chord AB.
Given The radius of the circle is 8 cm and the
distance to the chord from the centre of the
circle is 5 cm.
8
B
5
c2 a2 b2 b2 c2 - a2 82 - 52
39 b 6.2
A
Therefore, the length of the chord AB is 12.4 cm.
7.2.7
8
The Perpendicular Bisector of a Chord
In a circle of diameter 12 cm, two chords are on
the same side of centre. One chord is 10 cm and
the second 6 cm. How far apart are the chords?
Step 1 Find the distance to the
outer chord.
c2 a2 b2 62 a2 32 a2 36 - 9 27
a 5.2
6
6 cm
10
6
3 cm
Step 2 Find the distance to the
closer chord.
Therefore, the distance between the chords
is 5.2 - 3.3 1.9 cm.
c2 a2 b2 62 a2 52 a2 36 - 25 a 3.3
6 cm
5 cm
7.2.8
9
Assignment
Suggested Questions
Pages 400 and 401 1-20, 23, 25, 30
7.2.9
10
Chapter 7 The Circle
7.3A
Circle Theorems
1.
A line through the centre of a circle and the
midpoint of a chord is perpendicular to the
chord.
7.3A.1
MATHPOWERTM 11, WESTERN EDITION
11
Circle Theorems Contd
x
The measure of an inscribed angle is half the
measure of the central angle subtended by the
same arc.
2.
2x
3.
Inscribed angles subtended by the same arc are
congruent.
7.3A.2
12
Circle Theorems Contd
4.
An angle inscribed in a semi-circle is a right
angle.
5.
A tangent line is perpendicular to the radius
drawn to the point of tangency.
7.3A.3
13
Circle Theorems Contd
6.
Tangents drawn to a circle from the same
exterior point are congruent.
In a cyclic quadrilateral opposite angles
are supplementary.
7.
1
2
3
4
7.3A.4
14
Circle Theorems Contd
The angle between a tangent and a chord is equal
to the angle subtended on the opposite side of
the chord.
8.
1
2
4
3
9.
The measure of the central angle equals the
measure of the intercepted arc.
O
C
B
7.3A.5
15
Circle Theorems Contd
10.
The angle between the tangent and the chord is
equal to one-half the measure of the
intercepted arc.
2x
x
11.
B
The measure of the inscribed angle is one-half
the measure of the intercepted arc.
C
A
7.3A.6