CHAPTER 5 CIRCLE
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CHAPTER 5
CIRCLE
diameter
Circumference
radius
A Reminder about parts of the Circle
Parts
Definition: Circle – is a set of points each of which is equidistant from a fixed point called the center.
Circumference is the distance around the outer edge.
chord
Major Segment
Minor Segment
Minor Arc
Major Arc
A Reminder about parts of the Circle
Chord – line segment joining any two points on the circle.Arc – is a portion of a circle that contains two endpoints and all the points on the circle between the endpoints.Major Arc – longer arc Minor Arc – shorter arc
Minor Sector
Major Sector
A Reminder about parts of the Circle
A sector is the figure formed by two radii and an included arc.
A Reminder about parts of the Circle
A line is called a tangent line if it intersects the circle at exactly one point on the circle.A line is called a secant line if it intersects the circle at two points on the circle.
L2
P2
P1
L1
P
Exercises:
1. The area of a circle is 120 in2. What is its circumference? Ans. 38.83 in
2. A central angle of 136⁰ subtends an arc of 28.5 cm.What is the radius of the circle? Ans. r=12 cm
3. The angle of a sector is 30⁰ and the radius is 15 cm.What is the area of the sector in cm2? Ans. 58.90 cm2
The area of the segment is equal to the area of the sectorminus the area of the triangle formed by the two radii and
the central angle θ.A=(1/2)r2(θ - sin θ)
o
Arc AB subtends angle x at the centre.
AB
xo
Arc AB subtends angle y at the circumference.
yo
Chord AB also subtends angle x at the centre.Chord AB also subtends angle y at the circumference.
o
A
B
xo
yo
o
yo
xo
A
B
Introductory Terminology
Term’gy
Theorem 1
Measure the angles at the centre and circumference and make a conjecture.
xo
yo
xoyo
xo
yo
xo
yo
xo
yo
xo
yo
xo
yo
xo
yo
o o o o
o o o o
Th1
The angle subtended at the centre of a circle (by an arc or chord) is twice the angle subtended at the circumference by the same arc or chord. (angle at centre)
2xo
2xo 2xo 2xo
2xo 2xo 2xo 2xo
Theorem 1
Measure the angles at the centre and circumference and make a conjecture.
xo
xo
xoxo
xo xo xo xo
o oo o
o o o o
Angle x is subtended in the minor segment.
Watch for this one later.
o
AB
84o
xo
Example Questions
1
Find the unknown angles giving reasons for your answers.
o
AB
yo
2
35o
42o (Angle at the centre).
70o(Angle at the centre)
angle x = angle y =
(180 – 2 x 42) = 96o (Isos triangle/angle sum triangle). 48o (Angle at the centre)
angle x = angle y =
o
AB
42o
xo
Example Questions
3
Find the unknown angles giving reasons for your answers.
o
A
B
po
4
62o
yo
qo
124o (Angle at the centre)
(180 – 124)/2 = 280 (Isos triangle/angle sum triangle).
angle p = angle q =
o Diameter
90o angle in a semi-circle90o angle in a semi-circle20o angle sum triangle
90o angle in a semi-circle
o
a
b
c
70o
d
30o
e
Find the unknown angles below stating a reason.
angle a = angle b = angle c = angle d = angle e =
60o angle sum triangle
The angle in a semi-circle is a right angle.
Theorem 2
Th2
Angles subtended by an arc or chord in the same segment are equal.Theorem 3
xo xo
xo
xo
xo
yo
yo
Th3
38o xo
yo
30o
xo
yo
40o
Angles subtended by an arc or chord in the same segment are equal.
Theorem 3
Find the unknown angles in each case
Angle x = angle y = 38o Angle x = 30o
Angle y = 40o
The angle between a tangent and a radius is 90o. (Tan/rad)
Theorem 4
o
Th4
The angle between a tangent and a radius is 90o. (Tan/rad)
Theorem 4
o
180 – (90 + 36) = 54o Tan/rad and angle sum of triangle.
90o angle in a semi-circle60o angle sum triangle
angle x = angle y = angle z =
T
o
36oxo
yo
zo
30o
A
B
If OT is a radius and AB is a tangent, find the unknown angles, giving reasons for your answers.
The Alternate Segment Theorem.Theorem 5
The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment.
xo
xo
yo
yo
45o (Alt Seg)
60o (Alt Seg)
75o angle sum triangle
45o
xo
yo
60o
zo
Find the missing angles below giving reasons in each case.
angle x = angle y = angle z =
Th5
Cyclic Quadrilateral Theorem.Theorem 6
The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o)
w
x
y
z
Angles x + w = 180o
Angles y + z = 180o
q
p
r
s
Angles p + q = 180o
Angles r + s = 180o
Th6
180 – 85 = 95o (cyclic quad) 180 – 110 = 70o (cyclic quad)
Cyclic Quadrilateral Theorem.Theorem 6
The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o)
85o
110o
x y
70o
135op
r
q
Find the missing angles below
given reasons in each case.
angle x = angle y =
angle p = angle q = angle r =
180 – 135 = 45o (straight line) 180 – 70 = 110o (cyclic quad) 180 – 45 = 135o (cyclic quad)
Two Tangent Theorem.Theorem 7
From any point outside a circle only two tangents can be drawn and they are equal in length.
P
T
UQ
R
PT = PQ
P
T
U
Q
R
PT = PQ
Th7
90o (tan/rad)
Two Tangent Theorem.Theorem 7
From any point outside a circle only two tangents can be drawn and they are equal in length.
P T
QOxo
wo
98o
yo
zo
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons.
angle w = angle x = angle y = angle z =
90o (tan/rad)
49o (angle at centre)
360o – 278 = 82o
(quadrilateral)
90o (tan/rad)
Two Tangent Theorem.Theorem 7
From any point outside a circle only two tangents can be drawn and they are equal in length.
P T
QO
yo
50o
xo
80o
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons.
angle w = angle x = angle y = angle z =
180 – 140 = 40o (angles sum tri)50o (isos triangle)
50o (alt seg)
wo
zo
O
S T
3 cm
8 cm
Find length OS
OS = 5 cm (pythag triple: 3,4,5)
Chord Bisector Theorem.Theorem 8
A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord..
O
Th8
Angle SOT = 22o (symmetry/congruenncy)
Find angle x
O
S T
22o
xo
U
Angle x = 180 – 112 = 68o (angle sum triangle)
Chord Bisector Theorem.Theorem 8
A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord..
O
OS
T65o
P
R
U
Mixed Questions
PTR is a tangent line to the circle at T. Find angles SUT, SOT, OTS and OST.
Angle SUT =Angle SOT =Angle OTS =Angle OST =
65o (Alt seg)
130o (angle at centre)
25o (tan rad)
25o (isos triangle)Mixed Q 1
22o (cyclic quad)
68o (tan rad)
44o (isos triangle)
68o (alt seg)
Angle w =
Angle x =
Angle y =
Angle z =
O
w
y
48o
110o
U
Mixed Questions
PR and PQ are tangents to the circle. Find the missing angles giving reasons.
xz
P
Q
R
Mixed Q 2
•Extend AO to D•AO = BO = CO (radii of same circle) •Triangle AOB is isosceles(base angles equal)
D
•Triangle AOC is isosceles(base angles equal)
•Angle AOB = 180 - 2 (angle sum triangle) •Angle AOC = 180 - 2 (angle sum triangle) •Angle COB = 360 – (AOB + AOC)(<‘s at point) •Angle COB = 360 – (180 - 2 + 180 - 2) •Angle COB = 2 + 2 = 2(+ ) = 2 x < CAB
To prove that angle COB = 2 x angle CAB
QED
To Prove that the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference by the same arc or chord.
O
C
B
A
Theorem 1 and 2
Proof 1/2
Exercises:
4. Determine the area of the segment of a circle if thelength of the chord is 15 inches and located 5 inchesfrom the center of the circle. Ans. 42.32 in2
5. Find the area of the largest circle which can be cutfrom a square with an edge of 8 cm. What is the areaof the material wasted? Ans. 50.27 cm2 & 13.73 cm2
Surname, F.N. M.IStudent Number - Course
SUBJECT-SECTIONProfessor
Date
HW Format: (Portrait- Short Bond Paper)
1
2
Final AnswerFinal
Answer
3
4
1”
Write legibly & show your complete solution.
1”
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