CMV6120
Mathematics
Unit 8 : Angles Properties in Circles
Learning Objectives
The students should be able to:
recognize various parts of a circle. state the properties of chords of a circle. state and apply the property of angles at the centre. state and apply the property of angles in the same segment. recognize the property of angles in a semi-circle. explain the meaning of the concyclic points. state the properties of angles in a cyclic quadrilateral. state the definition of a tangent to a circle. recognize the properties of the tangents to a circle. state and apply the alternate segment theorem.
Unit 8: Angles properties in circles
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CMV6120
Mathematics
Circles
1. Parts of a circle
A circle is a closed curve in a plane such that all points on the curve are equidistant from a fixed point.
centre
The given distance is called the radius of the circle.
radius
A chord is a line segment with its end points on the circle and a diameter is a chord passing through the centre.
chord
diameter
An arc is a part of the circle. A segment is the region bounded by a chord and an arc of the circle.
major arc
major segment
minor segment minor arc
A sector is the region bounded by two radii and an arc.
sector
Unit 8: Angles properties in circles
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CMV6120
Mathematics
2.
Chords of a circle
Following are properties on chords of a circle. All these facts can be proved by the properties of congruent triangles. Theorem Theorem 1 The line joining the centre to the midpoint of a chord is perpendicular to the chord. i.e. If OM AB then MA = MB
P x M
Example O is the centre of the circle. Find the unknown in each of the following figures. 1.1 x = ________
O 4 cm
1.2 x =_________
Q
O x P N
Ref.: line from centre chord bisects chord Theorem 2 The line joining the centre of a circle and the mid-point of a chord is perpendicular to the chord. i.e. If MA = MB then OM AB 1.3
6 cm P T Q
r = ______
1.4
O
O
A M B
x2 =_________
x Q
3 cm
Ref.: line joining centre to mid-pt. of chord chord
M
8 cm
x= _________
Unit 8: Angles properties in circles
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CMV6120
Mathematics
Theorem Theorem 3 Equal chords are equidistant from the centre of a circle.
Example 2 O is the centre of the circle. Find the unknown(s) in each of the following figures. 2.1
i.e. If AB = CD, then OM = ON
M A O C N D B
4 cm M P 2 cm O R N 4 cmF S x cm Q
x = __________
2.2
R 2cm O 2 cm Q
Ref.: equal chords, equidistant from center
Theorem 4 Chords which are equidistant from the centre of a circle are equal. i.e. If OM = ON, then AB = CD
B
5 cm
P S y cm
y = __________
2.3
5 cm
Q 5 cm 3 cm R
O C N D
z w cm O
Ref.: chords equidistant from centre are eqaul
w = __________ z = __________
Unit 8: Angles properties in circles
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CMV6120
Mathematics
3.
Angles in a circle
C
As shown in the angle at by the arc ACB.
the the
figure, centre
O O A A C B B
AOB is subtended
ADB is the angle at the circumference subtended by the arc ACB
D B A
A C
B C
ADB is also called the angle in the segment ADB.
D B A
A C
Example 3.1 In each of the following figures, find the angles marked:a)
B x O A 78 C A y
b)
C 67 O B
Solution Theorem a) OA = OB b) Example 4 Join CO and product to D From a), y =
Unit 8: Angles properties in circles
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CMV6120
Mathematics
Theorem 5 (Angle at the centre theorem) The angle that an arc of a circle subtends at the centre is twice the angle that it subtends at any point on the remaining part of the circumference. i.e. If O is the centre of the circle, then AOB = 2 ACB
C
4.1
40 O P x
x = ______
4.2
4.3
D
P
B
Q 110
Ref.: at centre twice at
ce
4.4
O 180
Q R O P
R R
210
4.5
O x A 92 B
Theorem
Example 5
Unit 8: Angles properties in circles
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CMV6120
Mathematics
Theorem 6
O is the centre of the circle. Find the unknown(s) (Angles in a semi-circle theorem) in each of the following figures. 5.1
The angle in a semi-circle is a right angle. i.e. If AB is a diameter, then ACB=90.
C
84
O
x
x = __________
5.2 Ref.: in semi-circle Theorem 7 (Angle in the same segment theorem)
y x 3946
x = __________ y = __________
Angles in the same segment of a circle are equal. i.e. If ADB and ACB are in the same segment ABDC, then ADB = ACB 5.3
C D
O
38
x y
O
20
x Ref.: s in the same segment y =
4. Cyclic quadrilaterals 4.1 Concyclic points
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Unit 8: Angles properties in circles
CMV6120
Mathematics
Points are concyclic if they all lie on a circle, i.e. a circle can be drawn to pass through all of them.
An infinite number of circles can be drawn to pass through any two points.
If three points are not collinear, then one and only one circle can be drawn to pass through them.
If four points are concyclic, a circle can be drawn, but if they are not concyclic, no circle can be drawn to pass through all of them.
concyclic points
non-concyclic points
4.2
Cyclic quadrilateral
Unit 8: Angles properties in circles
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CMV6120
Mathematics
There are two important facts about a cyclic quadrilateral: i) A quadrilateral is called cyclic if a circle can be drawn to pass through all the four vertices. ii) All triangles are cyclic, but it is not true for quadrilateral.. Theorem Theorem 8 The opposite angles of a cyclic quadrilateral are supplementary. i.e. If P, Q, R, S are concyclic, then P + R = 180, and S + Q = 180
Q P
x 110
Example 6 O is the centre of the circle. Find the unknown(s) in each of the following figures 6.1
y O 85
x = __________ y = __________
Ref.: opp. s , cyclic quad. Theorem 9 If one side of a cyclic quadrilateral is extended, the exterior angle equals the interior opposite angle. i.e. If PQRS is a cyclic quadrilateral and PS is extended to T, then RST = PQR.
Q R
6.2
y
70 x
x = __________ y = __________
6.3
120
O y
82 P S T
x = __________ y = __________
Ref.: ext. , cyclic quad.
5. Tangents to a circle 5.1. Definition of a tangent to a circle
Unit 8: Angles properties in circles Page 9 of13
CMV6120
Mathematics
Figure 5.1 shows the three possibilities that a straight line (i) (ii) (iii)
Fig. 5.1
does not intersect a circle; intersects a circle at two points; touches a circle (i.e. intersects at one and only point).
(i) (ii) (iii)
When a straight line touches a circle, it is called a tangent to the circle at that point. The following theorem states a basic property of a tangent to a circle. Theorem 10 Example 7 AB is the tangent to the circle at T. Find the unknown
O
25
The tangent to a circle at a point is perpendicular 7.1 to the radius at that point. i.e. If TAB is a tangent at A, then OA TA
A O
a T 70 O C B
7.2
OTC =
B A T
c B
Ref.: tangent radius
7.3
O 46 b A T B C
OC = OT
5.2. Tangents from an external point to a circle
Unit 8: Angles properties in circles
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CMV6120
Mathematics
Theorem 11 If two tangents are drawn to a circle from an external point, a) the tangents are equal; b) the tangents subtend equal angles at the centre; c) the line joining the external point to the centre bisects the angle between the tangents.
Example 8 TA and TB are tangents to the circle at points A and B respectively. Find the unknowns. 8.1
A 5 cm O b 30
a
TA = a = b=
i.e. If TA, TB are tangents from T, then TA = TB; and TOA = TOB; and ATO = BTO
8.2
TA = TB c= d=
42
8.3
240
Ref.: tangent properties
TOB =
x
T
Unit 8: Angles properties in circles
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CMV6120
Mathematics
5.3. Alternate Segment Theorem
Theorem 12 (Alternate segment theorem) The angles between a tangent and a chord through the point of contact are equal respectively to the angles in the alternate segment. i.e. If then TAB is a tangent at A, TAD = ACD; and BAC = ADC Example 9 TB is a tangent to the circle at points A. Find the unknowns in each of the following figures.
9.1
46 50 O b A B
a =________ b = _______
a T
E D O C
9.2
c O d 45
c =________ d = _______
Ref.: in alt. Segment
9.3
35
y =
y x T B A
9.4 Z=
O z 30 B Unit 8: Angles properties in circles A
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CMV6120
Mathematics
Unit 8: Angles properties in circles
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