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CIRCLE THEOREMS. TANGENTS. A straight line can intersect a circle in three possible ways. It can be:. A TANGENT. A DIAMETER. A CHORD. B. O. O. O. B. A. A. A. 2 points of intersection. 2 points of intersection. 1 point of intersection. TANGENT PROPERTY 1. - PowerPoint PPT Presentation
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CIRCLE THEOREMS
TANGENTS
A straight line can intersect a circle in three possible ways.It can be:
A DIAMETER A CHORD A TANGENT
2 points of intersection
2 points of intersection
1 point of intersection
A
BO O O
A
B
A
TANGENT PROPERTY 1
O
The angle between a tangent and a radius is a right angle.
A
TANGENT PROPERTY 2
O
The two tangents drawn from a point P outside a circle are equal in length.
AP = BPA
P
B
ΙΙ
ΙΙ
O
A
BP
6 cm8 cm
AP is a tangent to the circle.a Calculate the length of OP.b Calculate the size of angle AOP.c Calculate the shaded area.
OP2 62 82
OP2 100
OP 10 cm
tan x
8
6
1 8tan
6x
53.13oAOP
c Shaded area = area of ΔOAP – area of sector OAB
a b
x
21 53.138 6 6
2 360
24 16.69
7.31 cm2 (3 s.f.)
Example
CHORDS AND SEGMENTS
major segment
minor segment
A straight line joining two points on the circumference of a circle is called a chord.
A chord divides a circle into two segments.
SYMMETRY PROPERTIES OF CHORDS 1
O
A B
The perpendicular line from the centre of a circle to a chord bisects the chord.
ΙΙΙΙ Note: Triangle AOB is isosceles.
SYMMETRY PROPERTIES OF CHORDS 2
O
A B
If two chords AB and CD are the same length then they will be the same perpendicular distance from the centre of the circle.
ΙΙΙΙ If AB = CD then OP = OQ.
C
D
ΙΙ
ΙΙ
P
Q
ΙΙ
AB = CD
O
96o
x
Find the value of x.
2x 96 180
2x 84
x 42o
Triangle OAB is isosceles because OA = OB (radii of circle)
Example
A
BSo angle OBA = x.
THEOREM 1
O
2x
x
The angle at the centre is twice the angle at the circumference.
O
96o
x
Find the value of x.
96 2x
x 96 2
x 48o
Angle at centre = 2 × angle at circumference
Example
O
62o
x
Find the value of x.
x 2 62
x 124o
Angle at centre = 2 × angle at circumference
Example
O
84o
x
Find the value of x.
84 2x
x 84 2
x 42o
Angle at centre = 2 × angle at circumference
Example
O
104o
x
Find the value of x.
y 2 104
y 208
Angle at centre = 2 × angle at circumference
y
x 360 208
x 152o
Example
THEOREM 2
O
An angle in a semi-circle is always a right angle.
O
Find the value of x.
x 58 90 180
x 32o
Angles in a semi-circle = 90o
and angles in a triangle add up to 180o.
58o x
Example
THEOREM 3
y
x
Opposite angles of a cyclic quadrilateral add up to 180o.
x y 180o
Find the values of x and y.
x 132 180
x 48o
Opposite angles in a cyclic quadrilateral add up to 180o.
x
y
75o 132o
y 75 180
y 105o
Example
THEOREM 4
x
Angles from the same arc in the same segment are equal.
x
x
39o
x
Find the value of x.
x 39o
Angles from the same arc in the same segment are equal.
Example
