New Century Mathematics (Second Edition) 4B

10 Basic Properties of Circles

Consolidation Exercise 10A

[In this exercise, unless otherwise stated, O is the centre of a circle.]

Level 1
In each of the following figures, AB is a chord of the circle. Find the unknowns. [Nos. 1–6]
1. A 2. A 3.
x cm 18 cm
a cm
M O
O N m cm
4 cm 6 cm
B O
A M B
B
16 cm

4. 5. M 6.
A A N B
x cm 2 cm b
30°
O O
3 cm O
N C y cm x
A B
P
B
N

ONC is a straight line.

In each of the following figures, PQ and RS meet at T. Find the radius of the circle. [Nos. 7–9]
7. P 8. P 9. Q

11 cm R C
C 4 cm
T 7 cm 3 cm
R S
T 3 cm
5 cm P
R T S
S
Q Q

C is the mid-point of PQ. C is the mid-point of PQ.

In each of the following figures, AB and CD are chords of the circle. Find the unknowns. [Nos. 10–12]
10. 11. 12.
15 cm B A
32 cm
B 4 cm D x cm
M
M D N
x cm
A M
O O
A O C
6 cm 10 cm N
10 cm a cm B
C N D
15 cm C

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13. In the figure, AB and PQ are two equal and parallel chords. MN A P
passes through O and OM ⊥ AB. If the distance between AB and PQ is
9 cm, find the length of OM.
M N
O

B Q

14. In the figure, AB = CD, OM ⊥ AB, BONE ⊥ CD, OB = 10 cm and B

NE = 4 cm. M

(a) Find the length of OM. A 10 cm

O
(b) Find the length of AB. C
N

E D
4 cm

15. In the figure, OP ⊥ AB, OQ ⊥ BC, OR ⊥ AC and OP = OQ = OR. A

Find ∠BAC.
P R
O

B Q C

16. In the figure, AED and BCD are straight lines. OM = ON, OM ⊥ AB,
E D
ON ⊥ AE and AB ⊥ BC. It is given that AB = 12 cm, BC = 4 cm and N
CD = 5 cm. A C
O
(a) Find the length of AE. M
B
(b) Find the length of DE.

17. In the figure, OQ and AB intersect at P. OQ ⊥ AB, OP = 15 cm and A

AB = 16 cm. 16 cm
Q P
(a) Find the radius of the circle. 15 cm
O
(b) Find the length of PQ. B

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18. In the figure, COD and ACB are straight lines. OC = 9 cm and D
AC = BC = 12 cm.
(a) Is OC the perpendicular bisector of AB? Explain your answer.
(b) Find the length of CD. O
9 cm
A B
C

12 cm 12 cm

Level 2
19. In the figure, C is the mid-point of AB. COD is a straight line and A
∠OBC = 28°.
(a) Find ∠BOC. O D
C
(b) Find ∠BDO.
28°

20. In the figure, AB and CD are two equal chords. OM ⊥ AB and

A C
ON ⊥ CD. If ∠MNO = 18°, find ∠MON.
O
M N
18°

B D

21. In the figure, PQ and RS meet at T. PR = PS, ∠RPT = ∠SPT and R

PQ = 24 cm.
(a) Show that !PRT ≅ !PST.
P Q
T
(b) Find the radius of the circle.

S
24 cm

22. In the figure, AMB and MON are straight lines. OM ⊥ AB, AM = 4 cm
4 cm M
A B
and OM = 3 cm.
3 cm
(a) Find the radius OA of the circle.
O
(b) Find the length of BN, correct to 1 decimal place.

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23. In the figure, ACB and MOCN are straight lines. OC ⊥ AB, AD // NM, M
D
AB = 12 cm and OC = 8 cm.
(a) Find the radius OA of the circle.
O
(b) Find the length of AD.
8 cm
C
A B
N
12 cm

24. In the figure, ABC and ODC are straight lines. CD = 14 cm,
A 24 cm B
AB = 24 cm and OA = 20 cm. Find the length of BC. C
20 cm 14 cm
D
O

25. In the figure, O is the centre of two concentric circles. AMBC and C
B
ANDE are straight lines. OM ⊥ AB, ON ⊥ AD and OM = ON. If
M
AB = 8 cm, BC = 3 cm and the radius of the smaller circle is 5 cm,
A O
find
N
(a) the length of AE, D
E
(b) the radius of the larger circle, correct to 1 decimal place.

26. In the figure, chords AB and CD are perpendicular to each other, and D
they meet at N. OM ⊥ AB, AB = 8 cm and OM = 3 cm. N is the mid- M
A B
N
point of MB.
(a) Find the radius of the circle. O
(b) Find the length of DN, correct to 1 decimal place.

27. In the figure, the chord AB and the radius OC meet at M, where 16 cm
OC ⊥ AB. AB = 16 cm and CM = 4 cm. Find the radius of the circle. C
4 cm
A B
M

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28. In the figure, AB and CD are two parallel chords. AB = 16 cm and
CD = 12 cm. The distance from O to CD is 8 cm.
(a) Find the radius of the circle. O

(b) Find the distance between AB and CD. 16 cm

A B
C D
12 cm

29. In the figure, OM ⊥ AB and ON ⊥ AC. ON = 2 cm, AC = 12 cm and

AD : DN = 3 : 1. C
(a) Show that !OND ~ !AMD. O
(b) Find the length of AB. N
D
M
A B

30. In the figure, OM ⊥ AB, ON ⊥ CD and MON is a straight line. If M

A B
AB = 32 cm, CD = 24 cm and MN = 28 cm, find the radius of the
circle. O

C N D

31. In the figure, P and Q are the centres of two circles. The two circles
B
intersect at A and B. M is a point on AB such that PM ⊥ AB. Does PM
produced pass through the centre Q? Explain your answer.
P M Q

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Answers
Consolidation Exercise 10A
1. 4 2. 9 20. 144°
3. 10 4. x = 5, y = 8 21. (b) 12 cm
5. 90° 6. 60° 22. (a) 5 cm (b) 8.9 cm
7. 8 cm 8. 7 cm 23. (a) 10 cm (b) 16 cm
9. 5 cm 10. 6 24. 18 cm
11. 32 12. 8 25. (a) 11 cm (b) 7.6 cm
13. 4.5 cm 26. (a) 5 cm (b) 1.6 cm
14. (a) 6 cm (b) 16 cm 27. 10 cm
15. 60° 28. (a) 10 cm (b) 2 cm
16. (a) 12 cm (b) 3 cm 29. (b) 7.2 cm
17. (a) 17 cm (b) 2 cm 30. 20 cm
18. (a) yes (b) 24 cm 31. yes
19. (a) 62° (b) 31°

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