test

Circle Properties 1

7515

The angle in a semicircle 1).

Draw a circle of any size, centre O. (You will be taking measurements from the diagram so make the radius of your circle greater than 5 cm).

Draw the diameter AB. At any point on the circumference mark the point C. Draw the lines AC and BC. Measure and record all the angles.

Repeat these instructions for another two circles. Write your findings.

2).

Copy each of the following diagrams and find the marked angles.

The letter O represents the centre of the circle. a).

b).

A

O

A

Diagrams not to scale

B

36° O

y°

y°

A

B

x°

C

d).

27°

x°

C

A

C

e).

B

f).

x°

C

y°

y°

O

O

O

O

68°

x°

C

c).

B

y°

A

B

49°

38°

B

C

O

C x°

y°

z°

A

A

x° B

g).

h).

C x°

y°

u°

B

B

42°

i).

A y°

v° O

w°

B

w°

A

x°

x° v° O

O 118°

w° 53°

y°

u°

v°

u°

C

C

A

j).

k). u° A O v° y° C

z°

Licensed to Sha Tin College

u°

A

A

v°

z°

v° y°

u° w°

B

70° y° O

x° w° B

l).

B

x°

O

w°

x°

C D

z° C www.10ticks.co.uk

The angle at the centre 1).

7516

Draw a circle of any size, centre O. Draw two radii OA and OB. At any point on the circumference mark the point C. Draw the lines AC and BC. Measure and record all the angles.

Repeat these instructions for another two circles. Write your findings.

C O

A

∠ ACB and ∠ AOB are said to be angles subtended by the arc AB. 2).

B

Copy each of the following diagrams and find the marked angles.

The letter O represents the centre of the circle.

A

a).

Diagrams not to scale c).

b).

y°

C 54° x°

O 98°

y° x°

B

O z° y°

A

O

x°

C

B

C

A A

d).

e).

f).

B

242

C

y°

D

C x°

O

x° ° O

A

2x°

127° B

O

x°

y°

y°

B

C

A A

C

g).

F

h).

z°

w° x° O y°

u° B

D E

A

j).

i). y° x° O

v° 158°

124°

E

147° D

w° v°

106°

E

C x°

B A

O w°

132° v° B

u°

u°

146° C

A

k).

B

E

A

u°

37°

C 111° D Licensed to Sha Tin College

x°

A

w°

O 63° w° B

D

l).

z°

v° w° O

B

49°

u° v° x° C

y°

O y°

D 57° B

D x° C www.10ticks.co.uk

Circle Properties 2

7517

D C

Angles in the same segment 1).

Draw a circle of any size.

Mark two points on the circumference A and B. Also on the circumference mark 3 more points C, D and E. Draw the lines as in the diagram. Measure and record all the angles.

Repeat these instructions for another two circles. Write your findings.

E

A

B

∠ ACB, ∠ ADB and ∠ AEB are said to be angles in the same segment.

2).

Copy each of the following diagrams and find the marked angles.

Where marked, the letter O represents the centre of the circle. a).

A

b).

B

64°

Diagrams not to scale

c).

B

w° 41°

D

u°

B

35°

D

A

A

e).

B

45°

E

u° v° 94° y°

A

E

f).

B

y°

D 142° A

v°

118° D x°

x° z°

w°

v°

w° y° O

C x°

53° C

D

A

k).

x°

D

Licensed to Sha Tin College

z°

w°

u°

v° D

O x°

74°

y°

y°

w°

C

D

B

B

l).

A

37° w°

C

u°

x°

B

v°

z°

86°

A

w° v° 2v°

A

z°

u°

B

y°

32°

A

v° O

A

j).

D

B F

i).

z°

u°

135°

C

h).

B

u°

C

C

g).

y°

w°

x°

D

D

v° x°

x°

A

47°

C

x°

d).

C

C

y° 32°

y° 44°

x° D

r°

u° 109° z° w° s° t°

B x°

v° C

C www.10ticks.co.uk

Angles in a Cyclic Quadrilateral.

1).

7518

Draw a circle of any size.

Draw a quadrilateral inside the circle as shown in the diagram.

Extend one of the sides of the quadrilateral.

Measure and record all the angles.

Repeat these instructions for another two circles. Write your findings.

A quadrilateral whose vertices all touch the circumference of a circle is called a cyclic quadrilateral.

Diagrams not to scale

Copy each of the following diagrams and find the marked angles.

Where marked, the letter O represents the centre of the circle.

2).

a).

A

115°

102°

b).

B

E

y°

w°

B

98°

w° C u° 72°

E

z° 164° D

v°

w°

O

O

B

60°

E D u° 100° v°

k).

F

q°

y°

w°

p° 43°

z°

y° z°

32°

u° v° w°

x° A

t° x° D 85° E

Licensed to Sha Tin College

r°

F

s°

u° D 133°

l).

E

B D

O

64°

C C

w° B v°

z° C

E

A 99°

B

A x° y°

v° C 100° x° u° D y°

B

w°

36°

71°

v°

j).

C

i).

w° A

A

2u° 3u°

x°

E

C

E

x°

232° u° O

D

y°

h).

D

C

A

x° A

x°

f).

B

78° v°

B

v° C

F

C

u°

121° x°

108° B

e).

B

g).

A

D

E

d).

D

F

C

D

A

c).

x°

83°

C

x°

A

D z° y°

z°

y°

A

x° B www.10ticks.co.uk

Circle Properties 3

7519

B

Tangents 1).

Draw a circle of any size.

A Draw two tangents to the circle that meet at A. Where the tangents touch the circle join them to the centre, O. Measure and record all the angles and lengths.

Repeat these instructions for another two circles. Write your findings.

O

C

A tangent touches a circle. It does not cross the circumference.

2).

Copy each of the following diagrams and find the marked angles and lengths.

The letter O represents the centre of the circle and the lines AB and AC are tangents.

a).

A

b).

c).

12 cm

B

y cm

x° B

y°

37°

A

Diagrams not to scale

O

18 cm

O

x° 5 cm

y cm

O

A

12 cm

x° B

d).

e).

f).

B

B x°

6m

x°

z°

ym

A 28°

O

D

z°

A

y°

O A

D

y°

56°

O

x° B

g).

h).

B

v°

v°

O

56° w° O

C

i).

u°

A

w° D

x°

B A

v° x°

x°

D

y° O w°

z°

69°

u°

u°

130° D

B

y°

j).

43 mm A

x°

y° z mm

A

k).

B

D

v°

w° 63° u° C

O

l).

B r°

O

x cm

w° O

s° u°

38° v° t° C

19 cm

A

B

u°

s° p° t° q°

r°

C

D v°

A

Licensed to Sha Tin College

www.10ticks.co.uk

Circle Properties 4

7521

Mixed Questions

Copy each of the following diagrams and find the marked angles and lengths.

The letter O represents the centre of the circle.

Diagrams not to scale

A

1).

2).

A

3).

9 cm

C

y cm

47°

y°

y° x°

O B

B

O

x° 4 cm O

x°

73°

B

A

C

4).

E

5).

D z° y°

6).

B

A x°

y°

112° 86°

O 34°

x°

B

46°

D

A

8).

A

w°

9).

A

B

52°

u° v° 101° y°

236

12).

F

z° x°

D

x° B

B O y°

O

A

z°

B

11).

63°

O

y°

C

y°

C

10).

C

x°

° O

x°

D

A

x°

C

A

C

7).

C

B

D 117° x°

A u°

s° q° B

u° t° p° D v°

r°

C

v° w° C 93°

B

E

A

Licensed to Sha Tin College

www.10ticks.co.uk

E

13).

14).

C u°

x°

D 145°

B

39°

A

v°

w°

O w°

x°

v°

z°

O

y°

v°

B

y°

128° D

u°

C

u°

x°

w°

A

7522

15).

y°

131° B

A

F

16).

17).

D

18).

A 27°

A

119° B

x°

w°

y°

O

3v° x°

D

B

B

y°

y° w°

z°

A

C

v° y°

u° w°

21).

D

84°

w°

t°

23).

A

C

v°

B

C

z°

t°

r°

26).

y° x° O

E

24).

A

B

u°

w° v°

27).

B

w° B A x° v° y°

z mm

102° B

A

u°

139°

Licensed to Sha Tin College

D

C

49°

C

y°

v° x°

u° D

C

z°

A

z°

E

v°

D

58° w°

x° 36°

B

A z°

x°

118° w° z° s°

F

25).

B

u°

O y°

w°

x°

E

62°

C

w°

236° u° O

D

x cm

D u° 101° v°

33°

A

C

C

22).

39° v°

47°

z°

D

y°

9.4 cm

s° u°

O

O x°

A

B r°

u°

A

20).

B

104°

x° v°

C

19).

O

2v°

C

x°

v°

y° 47 mm

w° O 59°

u°

C

D 136° E www.10ticks.co.uk

Trigonometry and Circle Properties 1).

The points A, B and C lie on the circle, centre O.

BC is a straight line.

If AB = 4.5 cm and AC = 7.3 cm, find

a). ∠BAC,

b). ∠BCA,

c). the radius of the circle.

AB is a tangent to the circle, centre O. AO is a straight line passing through the circle at D. If AO = 145 mm and AB = 116 mm, find a).

OB, b). ∠BOD, c). ∠OBD, d). ∠ADB.

4).

AB and AC are tangents to the circle, centre O. AO is a straight line passing through BC at D. If AO = 10.8 cm and OC = 4.3 cm, find a).

AC, b).

AB, c). ∠BAO, d). ∠AOB, e).

OD, f).

BD, g). ∠OBD, h). ∠DBA.

5).

The points A and B lie on the circle, centre O. AC = BC. If ∠OAC = 52° and AC = 6.3 cm, find a).

AO, b).

CB, c).

OC, d). ∠AOC, e). ∠AOB.

6).

The points A and B lie on the circle, centre O. If ∠AOB = 122° and AO = 148 mm, find a).

AB, b). ∠OAB. Hint: You may have to draw some other lines.

Diagrams not to scale

B O C A

2).

The points A, B and C lie on the circle, centre O.

BC is a straight line. OB = OC = OA. ∠AOC = 122°.

If OB = 13 cm, find

a).

AC,

b). ∠ACO,

c). ∠ABC,

d). ∠CAO. 3).

7538

B O C A

D

A

O

B

B

D

A

O

C

O

A

C

B

A

O

B Licensed to Sha Tin College

www.10ticks.co.uk