Chapter 7. Circle

Chapter 7

CIRCLE

Standard Competence :6. Identifying the circle and finding the measure of its parts.

Base Competence :6.1. Recognize the circle and its part.6.2. Finding the measure of parts of the circle.

Indicators :a. Differentiate the circle and the area of circle and name the part of the circle ( center, radius,

diameter, arc, sector of circle, segment of circle )b. Determining the value of ( pi )c. Drawing the Circumscribed circle and Inscribed circle of a triangle and drawing a circle

passing through three point are given.d. Finding the circumference and the area of a circlee. Finding the change of the area of circle if its radius is change.f. Finding the length of arc, the area of sector and segment of the circle.g. Recognize the relation of center angle and inscribed angle in the same opposite arc.h. Finding the inscribed angle if the opposite diameter and opposite arc are same.

A. THE CIRCLE AND SOME PARTS OF IT

1. THE CIRCLE AND THE AREA OF CIRCLELook at the following picture.

Circle The Area of circle ( the shaded region )

2. SOME PARTS OF CIRCLE.

D O = The center of circle E CF = Diameter

OB,OC,OD,OF = RadiusOH = Apothem

F H O C CD,EG = Chord CD, BC = Arc G B Segment CD

Sector OBCSegment is part of the circle bounded by a chord and an arc.Sector is part of the circle bounded by two radii and an arc

Students Worksheet – Mathematics for 8th RSBI class - 1 -

B. APPROXIMATE THE VALUE OF ( PI )

The ratio of the circumference of any circle to its diameter is always a constant number , then called (pi). The value of (pi) = 3.1415926536… ( an Irrational Number )Look at the following table

Diameter ( d ) Circumference ( C )Cd ( )

5 cm 15.7 cm 3.148 cm 25.12 cm 3.1410 cm 31.4 cm 3.1412 cm 37.68 cm 3.14

Consider the table above, then we can find the approximate value of .Approximate the value of = 3.14 ( round off to 2 decimal place )

=

227

C. DRAWING INSCRIBED AND CIRCUMSCRIBED CIRCLE OF A TRIANGLE.

1. INSCRIBED CIRCLE OF THE TRIANGLE

C 1. Draw bisector of the angle A

bisector (with A as center, draw arc the intersection both the angle A sides at D and E. with D as center and with E as

center draw arc in interior angle in the same radi- E us, then its intersectioa at F. Connect A and F)

F 2. Draw bisector the angle B ( same the method 1 )3. The center of Inscribed Circle of triangle is the in-

A D B tersection point of the bisector angles of triangle.

2. CIRCUMSCRIBED CIRCLE OF THE TRIANGLE.

C 1. Draw the perpendicular bisector of side AB ( with A as center and opning the radius greater

than half of AB, draw a semicircular arc. With Bas center and same the first radius, then draw a semicircular too. Connect two points of intersection.

Its line is called the perpendicular bisector ) A B 2. Draw the perpendicular bisector of side AC.

( same the method 1 )3. The Centre of Circumscribed circle of triangle is

the point of intersection of the perpendicular bisector of triangle.


3. THE LENGTH OF THE RADIUS OF INSCRIBED CIRCLE OF THE TRIANGLE

CThe length of radius of Inscribed Circle

r O

A B Note :

r = the radius of Inscribed circleA = the Area of triangle

s =

the Circumference of Circle2 =

12(

a+b+c )

4. THE LENGTH OF THE RADIUS OF CIRCUMSCRIBED CIRCLE .

C The length of radius of Circumscribed Circle P

O

A B

Note :

r = the radius of Circumscribed circleA = the Area of trianglea, b, c = the sides of triangle

TASK 1

Complete :

1. Name the the following some parts of the circle below.

A O = …. E OA,OB, OC = ….

B AB = ….BD = ….

D O OF = …. F AG = ….

BD = …. G ….

C ….….


r =

As

CD, AB = ….

2. The value of ( pi ) = ….= … ( round off to 5 decimal place )= … ( round off to 4 decimal place )= … ( round off to 3 decimal place )= … ( round off to 2 decimal place )

3. Given the Δ ABC.

Proof that the length of radius of Inscribed circle of the Δ ABC is r =

As

Proof :

C The Area of Δ AOB =

12

.. . . x r

The Area of Δ BOC =

12

.. . . x r

r The Area of Δ COA =

12

.. . . x r

A B +

The Area of Δ ABC =

12

r ( . . . + . . . + . .. )

The Area of triangle ABC. .. . = r =

4. Given the Δ ABC.

Proof that the length of radius of Circumcribed circle of the Δ ABC is r =

abc4 A

Proof :Look at Δ BTC and Δ APC

C ACP = …. ( 90o )

P P = … (

12

AC)

O Δ BTC and Δ APC are similar

A T B Then,

AP. ..

= AC.. ..

AP =

. .. . x . .. .

. .. .

Look at the ABC 2 r =

. .. x .. .

. ..

A =

12

.. . . xTC 2 r =

. .. x .. .

. ..


r =

A. .. .

X 2

…. A = … x TC

TC =

. .. A

. .. .

5. Given the PQR, PQ = 5 cm, QR = 4 cm and RP = 3 cm.Find : a. the radius of Inscribed Circle of PQR

b. the radius of Circumcribed Circle of PQRSolution :

a. p = 4 cm S =

12( . .. .+. .. .+.. .. )

r =

As

q = 3 cm = …. cm =

. .. .

. .. .r = 5 cm r = ….

Area =

12

bh

4 cm 5 =

12

. .. . . .. . .

= ….

3.

b.

r =pqr4 A

=.. . . x . .. . x . . ..4 x . .. .

= . . ..

D. THE CIRCUMFERENCE AND THE AREA OF CIRCLE.

1. THE CIRCUMFERENCE OF A CIRCLE

The ratio of the circumference of any circle to its diameter is always constant.Then is called ( pi ).

Cd =

C = d or C = . 2r C = 2 r

Note :C = the Circuferencer = radiusd = diameter

= 3.14 or

227


r =

. .. .

. .. .

C = d or C = 2 r

2. THE AREA OF A CIRCLE .

Take the circular carton and draw a circle with 16 sectors, then arrange like the following figure.

about

12

C = r

O R P S

P Q п r

The Area of Rectangle PQRO = l . w= r . r= r 2

The Area of Circle = The Area of Rectangle PQRO= r2

TASK 2

1. Find the circumference of the wheel whose diameter are :

a. 14 cm c. 2√5 cm e. 2 п cm

b. 20 cm d. 17.5 cm f. √3 cm

2. Find the cover of long arm on o`clock for one minute if the radius are :a. 100 mm c. 3.5 cm e. 3 п cm

b. 25 mm d. 28 mm f. 2√6 cm

3. Complete the following table.

No Radius ( r ) Diameter ( d ) Circumference( C ) Area ( A )

i … 14 cm … …


A circle = r 2

A circle =

14 d 2

ii 40 cm … … …iii … … 62.8 cm …

iv … … …12

47 cm2

v … … … 28.26 cm2

4. The diameter of motorcycle wheel is 42 cm.Find the distance traveled if its wheel rotates for 500 times.

5. The diameter of flat zinc is 17.5 cm. If we make 4 circular holes on it and the diameter each hole is 7 cm. Find the area of the zinc outside the hole

6. Find the area of the circles whose :a. radius = 5 cm c. radius = 14 cmb. diameter = 10 cm d. diameter = 21 cm

7. Find the area of the circles whose circumferences are :a. 44 cm c. 6 п cmb. 154 cm d. 314 cm

8. Find the radius of Inscribed circle and circumscribed circle of :a. the triangle ABC, AB = BC = 10 cm and AC = 13 cmb. the square ABCD, AB = 14 cm

9. The radii of two circle are 4 cm and 6 cm. What is the ratio of :a. their circumferenceb. their area

9. Find the circumference and the area of the following shaded region :

a. b. c.

28 cm 20 cm 14 cm


E. THE CHANGE OF THE AREA OF CIRCLE

Example :1. The area of the circle A is 616 cm2. Find the area of circle B if its radius twice of the

radius of the circle A.

Solution :Circle A Circle B

A = п r 2 = 616 A = п r 2

227

r2

= 616 =

227 x 28x 28

r 2 = 616 :

227 = 88 x 28

r 2 = 616 x

722 = 2464 cm 2

r = √196 r = 14 cm d = 2 x 14

d = 28 cm

F. THE LENGTH OF ARC, THE AREA OF SECTOR AND SEGMENT OF CIRCLE.

1. THE LENGTH OF ARC AND THE AREA OF SECTOR.

The length of Arc, the area of circle,circumference and the area of circle have the relationIf a circle is cut into 4 congruent sector, what is the length of Arc AB ?Find the Area of AOB !

BWe can use the ratio :

O

90o

360o= Arc AB

Ccircle

= Area of sec tor AOBAcircle

C A

D


α o

360o= Arc AB

Ccircle

= Area of sec tor AOBAcircle

The Area of segment AB = Asector AOB - A AOB

= the center angle of sector , then :

2. THE AREA OF SEGMENT.

B

A O

TASK 31. Consider the figure on the left.

The radius of circles A and B are 3.5 cm and 7 cm.If the wheel A rotates for 100 times, how many times

A B does the wheel B rotates ?

2. The radii of two circles are 9 cm and 4 cm. What is the ratio of their Area.

3. The diameter of a circle is 28 cm. Find the length of arc AC if its center angle are :a. 30o c. 200o

b. 45o d. 22.5o

4. Find the area of the sector if Its radius and the center angle are :a. 21 cm and 60 o

b. 10 cm and 150 o

5. The radius of a circle is 10 cm and the length of chord PQ is 10 cm, find the area of segment PQ.

G. THE RELATION OF CENTRE ANGLE AND INSCRIBED ANGLE.

C BOC is the center angle BAC is the inscribed angle


The length of Arc AB =

α360

x Ccircle

The length of Arc AB =

α360

x Acircle

x (at the circumference) A x B O

The center angle is twice of Inscribed angle, if C in the same opposite arc

B 40 o 80 o

80 o AC = 80 o

A

H. THE ANGLE BETWEEN TWO CHORDS.1. THE INTERNAL INTERSECTION OF CHORDS

D C APD = the Internal Intersection of chords AC and BD AD x A P BC

B

2. THE EXTERNAL INTERSECTION OF CHORDS.

T APD = the Externmal Intersection of chords AC and BD

S

x P

Q R

I. CYCLIC QUADRILATERAL.

Quadrilateral ABCD is Cyclic Quadrilateral D All of sides of Cyclic Quadrilateral are Chords.

C

B =

12

∩ AC ( AC minor )

A B D =

12

∩ ACmajor


BOC = 2 x BAC

B + D = 180o

The sum of opposite angle on Cyclic Quadrilateral is 180o

TASK 4

1. Given a circle whose radius 14 cm, AOB = 72 o . Find :a. the area of sector AOBb. the length of Chord AB

2. Find x, y and za. b. c.

C

x O x B 50o y A O C

y z 30o

35o x

A

3. Look at the circle on the left.y Find x, y and z

x z

35 o

4. N Given KML = 48o and arc MN = 60

M Find x

x L

K

5. E Given, AOE = 150o and BAD = 40o

Find ACE D

O C

A BStudents Worksheet – Mathematics for 8th RSBI class

- 11 -

6. D C Given the circle on the left.AE = 8 cm, EC = 3 cm and BE = 6 cm.Find DE

E

A B

For no. 7 – 10 Find x, y and z

7. 8. C

y 5x y OAB = 25o

ABO = x and 80o O AOB = z

3x A B

9. 10. D D

z y C E

30o x 50o

A B A B C

AD // BC AC BD

DAC = 60o , ADB = x, ACB = y, CBD = z


COMPETENCE TEST

I. Choose the right answer by crossing (x) on a, b, c or d.

1. The length of arc AB is ...

a. 5.1 cm O b. 7.3 cm

14 cm c. 10.2 cmd. 14.6 cm

B A

2. A Given AOB = 78o

C B Then ACB = ….a. 12 o

b. 39 o

O c. 112 o

d. 156 o

3. D C Look the circle on the left. DEC = ….

E a.

12∠ AEB

A B b.

12∠ ACB

c. AEB d. ADB

4. The area of a circle is 376.8 cm2. AOB= 30o. The area of sector AOB is ….

a. 125.6 cm2 c. 31.4 cm2

b. 62.8 cm2 d. 12.56 cm2

5. Given OQ = 6 cmThe area of segment

AB B is ….

O a. (12 - 9√3 ) cm

b. (12 - 6√3 ) cm

c. ( 6 - 6√3 ) cm

d. ( 6 - 9√3 ) cm6. B

C Given the length arc AB

= 18 cm, then the length 45o 60o A arc BC = ….

O a. 12 cmb. 14 cmc. 15 cmd. 16 cm

7. B ABC = ….

a. 55o

250 b. 65 o

c. 105 o

C d. 125 o

AStudents Worksheet – Mathematics for 8th RSBI class

- 13 -

8. The length of arc PQ = 2.5 cm. If AOB=45o ,then the circumference of circle is…a. 20 cm c. 26 cmb. 24 cm d. 28 cm

9. Given AC BD D The angle of arc BC is A …

a. 80 o

30 b. 91 o

c. 120 o

B C d. 135 o

10. Given AOD = 120 BAC = 25o, then

D ABD = …. C a. 25 o

O B b. 30 o

c. 35 o

A d. 60 o

11. D Consider the circle A on the left, then O C ADC = …

a. 60 o

b. 110 o

60 c. 120 o

B d. 210 o

12. ABC = … C a. 70 o

b. 90 o

c. 110 o

100 d. 140 o

120 A B13. The radii of circle A and B are 3cm and

4cm. The ratio of the area of the circles is …a. 3 : 4 c. 9 : 16b. 6 : 8 d. 27 : 64

14. Given the triangle KLM, KL = 9 cm, LM =12 cm and KM = 15 cm. The length of radius of the circumscribed circle of KLM is ….

a. 4.5 cm c. 7.5 cmb. 7 cm d. 9 cm

15. Given the circumference of a circle is 88 cm and AOB = 45o. The area of sector AOB is ….a. 11 cm 2 c. 154 cm 2

b. 77 cm 2 d. 616 cm 2

16. The cover of a motorcycle is 18,840 cm if its wheel rotate 100 times, then the radius of the motorcycle wheel is ….a. 20 cm c. 40 cmb. 30 cm d. 50 cm

17. The circumference of a circle whose diameter 21 cm is ….a. 66 cm c. 198 cmb. 132 cm d. 264 cm

18. Given COB = 60o, B then OBA is ….

C a. 10 o

b. 20 o

O c. 25 o

d. 30 o

A

19. The angle of arc AD is ... C

a. 30 o

b. 70 o

D c. 80 o

70o

30o d. 140

o

B A E

20. Given a circle on the left. The perimeter of

A sector AOB is …. O 21 cm B a. 100 cm

b. 97 cmc. 95 cmd. 90 cm


II. Do the following problems.

1. A circle whose radius 10.5 cm. Find :a. its circumferenceb. its area.

2. The diameter of a circular field is 100 m. Oktapriana run 6.28 km around it. How many times did he run around it.

3. Given APB + AQB + ARB = 144 o A B Find :

a. APB P b. AOB O R

Q

4. D C Consider the figure on the left.a. find the area of shaded regionb. find the circumference of shaded region

A 14 cm B

5. Given the circle ADE = 60 o , DEB = 30o and BDE = 110

E Find : D a. BCD b. EFA

O F C B

A


Practice makes perfect


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Chapter 7. Circle