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Units_007-090 FV
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Angles
Circle geometry
1
Key facts
• The parts of a circle are shown in these diagrams.
(a)
angle at circumference
on minor arc
(b)
segment
chord
eter
diam
umference
circ
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angle at centre
radius
tor
sec
tangent
arc
point of contact
angle on major arc
• There are four basic circle geometry facts to which others are linked.
Fact 1 A tangent to a circle is perpendicular to the radius at the point of
contact. (This can be proved by contradiction.)
S
C
P
T
Deduction 1 The 2 tangents from a point to a
x
circle are equal. Triangles congruent (SSS).
Fact 2 An angle at the centre of a circle is equal
2y
2x
to twice any angle at the circumference which
stands on the same arc.
y
Deduction 2
(a) All angles at the circumference which stand
on the same arc are equal.
(b) The angle in a semicircle is a right-angle.
(c) Opposite angles of a cyclic quadrilateral are supplementary.
(d) The exterior angle of a cyclic quadrilateral equals the interior opposite
angle.
(a)
(b)
(c)
(d)
a
b
d
b
c
a
d
c
a + c = 180°
b + d = 180°
e
e=b
continued
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Fact 3 The angle between a chord and tangent equals any angle in the
alternate segment.
x
x
Fact 4 A line from the centre of a circle perpendicular to the chord bisects
the chord
Worked example
Example 1
Example 2
Given that TQ = TR calculate the sizes of the
In this diagram C is the centre of the circle, PT is
angles with letters.
a tangent to the circle and M is the midpoint of
Give reasons.
chord PA. Name all the right-angles in the
diagram. Give reasons.
R
Q
c
P
B
a
b
S
d
C
70°
T
A
T
M
P
Solution 1
a = 70°
(Angle in the alternate segment.)
Solution 2
b = 70°
(Base angles of isosceles
angle CPT = 90°
triangle.)
c = 110°
d = 70°
(Tangent is perpendicular
to the radius.)
(Opposite angles of cyclic
angle PAB = 90°
quadrilateral.)
angle CMA = 90°
(Angles in the same segment.)
and angle CMP = 90° (CM bisects and so is
(Angle in a semicircle.)
perpendicular to chord
AP.)
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Definitions
• Tangent a straight
line which touches
a circle at a point of
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! Examiner’s tips
•
•
Do not assume that lines which cross inside a circle do so at the centre
unless this is made clear in the question.
• Remembering the angle in a semicircle is important but be careful not to
assume angles at the circumference are 90° unless it is clear they stand
contact on the
circumference
• Cyclic quadrilateral
a quadrilateral with
on a diameter.
• Turning diagrams upside down may help in looking for related angles.
• Add the sizes of angles to the diagram as you find them.
all its vertices on
the circumference
Can you answer these questions?
of a circle.
1. Find the angles marked with letters in these diagrams. Give reasons.
(a)
Did you know?
•
The word ‘tangent’
comes from the
Latin word
‘tangere’, meaning
‘to touch’, which is
also the root of
‘tangible’, meaning
‘can be touched’.
(b)
70°
83°
(c)
(d)
a
b
78°
C
d
62°
s
x
r
f
e
C is the centre
2. A point P is 10 cm from a point C which is
the centre of a circle radius 4 cm.
Calculate the length of the tangent to the
y
x
circle from P. (You will need to sketch a
Look on
the CD
for more
exam
practice
questions
58°
diagram.)
a
3. Prove fact 2 by using the isosceles
b
triangles in this diagram to find
connections between angles a and x and
between y and b.
4.
c
a
d
O is the centre of the circle
and PT is a tangent
O
T
b
e
P
(a) Write down the value of a. Give a reason.
(b) Find e, c and d in terms of b. Give reasons.
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