Chapter5 Circle

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”