Chapter 7 Quadrilaterals

1

Chapter 7 Quadrilaterals

7A p.2

7B p.16

7C p.30

7D p.37

Chapter 8 More about 3-D Figures

8A p.49

8B p.57

8C p.66

8D p.77

Chapter 9 Area and Volume (III)

9A p.89

9B p.104

9C p.120

9D p.131

9E p.137

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2

F3B: Chapter 7A

Date Task Progress

Lesson Worksheet

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(Full Solution)

Book Example 1

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Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Consolidation Exercise


(Full Solution)

3

Maths Corner Exercise

7A Level 1


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7A Level 2


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7A Level 3


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7A Multiple Choice

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E-Class Multiple Choice

Self-Test

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4

Book 3B Lesson Worksheet 7A (Refer to §7.2)

7.2A Properties of Parallelograms

(a) Definition of a parallelogram:

a quadrilateral with two pairs of parallel opposite sides

(b) Properties of parallelograms:

(i) The opposite sides of a parallelogram are equal.

i.e. In the figure, AB = DC and AD = BC.

[Reference: opp. sides of //gram]

(ii) The opposite angles of a parallelogram are equal.

i.e. In the figure, ∠A = ∠C and ∠B = ∠D.

[Reference: opp. ∠s of //gram]

(iii) The diagonals of a parallelogram bisect each other.

i.e. In the figure, AE = EC and BE = ED.

[Reference: diagonals of //gram]

1. Fill in the blanks for each of the following parallelograms.

(a) (b)

PTR and STQ are straight lines.

(c) (d)

WMY and XMZ are straight lines.

A B

C D

7

48°

Q R

S P

4 9 T

52°

7

E F

G H 10

W X

Y Z

M

9

5 6

A B

C D

A B

C D

E

A B

C D

The opposite side of DC is ____.

The opposite angle of ∠A is ∠__.

The diagonals PR and SQ bisect each

other.

i.e. PT = ____ and ST = ____.

5

○○○○→→→→ Ex 7A 1−3

Example 1 Instant Drill 1 In the figure, ABCD is a parallelogram. AC and

BD intersect at E. Find the unknowns.

Sol ∠DCB = ∠DAB (opp. ∠s of //gram)

x = 25° + 27°

= 52° DE = EB (diagonals of //gram)

2y = y + 6

y = 6

In the figure, RUST is a

parallelogram. RS and

TU intersect at O. Find

the unknowns.

Sol RT = ____ ( of //gram)

h = ___________

=

Find the unknowns in each of the following parallelograms. [Nos. 2−−−−5]

2.

PTR and QTS are straight lines.

3.

4.

BEC is a straight line.

5.

CEB is a straight line.

○○○○→→→→ Ex 7A 4−9

Q R

S P

2x

x + y 10 9 T

Q R

S P

y

2x − 3°

x + 34°

E A

B

C

D

p q

25° 21°

A

B

C

D

q 3p

2p E

60°

AD // BC

∵ PQ // SR ∴ ∠Q + ∠R

= ______

A

B C

D

2y

y + 6 x

27°

25°

E

R

U

S

T

k

h

h − 2

3h − 14

O

6

Example 2 Instant Drill 2

In the figure, PQRS is a parallelogram. QTR is

a straight line. Find the unknowns.

Sol ∠PTQ + 112° = 180° (adj. ∠s on st. line)

∠PTQ = 68°

In △PQT,

∵ PQ = PT (given)

∴ ∠PQT = ∠PTQ (base ∠s, isos. △)

x = 68°

∠S = ∠Q (opp. ∠s of //gram)

y = x

= 68°

In the figure, ABCD is a parallelogram. Find

the unknowns.

Sol ∠DAB = ∠______ ( )

=

In △ABD,

∵ BD = ____ ( )

∴

6. In the figure, CEFG is a parallelogram. CGH and DEF are

straight lines. Find x.

○○○○→→→→ Ex 7A 10−12

7. In the figure, QTR is a straight line. PSRT is a parallelogram.

If PS = 9 cm and QR = 15 cm, find

(a) the length of PT,

(b) the perimeter of quadrilateral PQRS.

○○○○→→→→ Ex 7A 13, 14

Q R

S P y

112° x

T

B C

D A

y

70° x

Is △FGH isosceles? C

E F

G H

x

D 75°

Q R

S P

T

△PQT is an

__________

triangle.

7

7.2B Tests for Parallelograms

Conditions for identifying a parallelogram:

(a) Both pairs of opposite sides of a quadrilateral are equal.

i.e. In the figure, if AB = DC and AD = BC,

then ABCD is a parallelogram.

[Reference: opp. sides equal]

(b) Both pairs of opposite angles of a quadrilateral are equal.

i.e. In the figure, if ∠A = ∠C and ∠B = ∠D,


[Reference: opp. ∠s equal]

(c) The diagonals of a quadrilateral bisect each other.

i.e. In the figure, if AE = EC and BE = ED,


[Reference: diags. bisect each other]

(d) One pair of opposite sides of a quadrilateral are equal

and parallel.

i.e. In the figure, if AB // DC and AB = DC,


[Reference: 2 sides equal and //]

8. Determine whether each of the following quadrilaterals must be a parallelogram. If yes, write

down the reasons.

(a) (b)

� Yes, ____________________ � Yes, ____________________

� No � No

(c) (d)

� Yes, ____________________ � Yes, ____________________

� No � No

(e) (f)

� Yes, ____________________ � Yes, ____________________

A

B C

D

135°

45°

45°

135°

E

F G

H 8

8

L

K J

I 10

10

6 6

U T

R S

12

12

8

8

K

L M

N

4 4

P

Q

R

S 11

11

A B

C D

A B

C D

A B

C D

A B

C D

E

8

� No � No

Example 3 Instant Drill 3 In the figure, AC and BD intersect at E. Prove

that ABCD is a parallelogram.

Refer to the figure. Prove that EFGH is a

parallelogram.

Sol AE = EC = 7

BE = ED = 6

∴ ABCD is a

parallelogram.

given

given

diags. bisect each

other

Sol EF = ____ = 3

EH = ____ = ____

∴ ____________

____________

given

given

___________________

___________________

9. Refer to the figure. Prove that PQRS is a

parallelogram.

10. Refer to the figure. Prove that ABCD is a

parallelogram.

11. In the figure, PST is a straight line. Prove that PQRS is

a parallelogram.

○○○○→→→→ Ex 7A 15

A

B C

D

E

6 6

7 7

E

F G

H 4 4

3 3

P

Q

R

S 80°

80°

100°

100°

A

B C

D 10 10 133° 47°

Is AB parallel

to DC?

P

Q

R

S

T

We can also identify a

parallelogram by

its definition (see

P.7A-1).

9


Refer to the figure.

(a) Find x.

(b) Prove that ABCD

is a parallelogram.


(a) Find a.

(b) Prove that PQRS

is a parallelogram.

Sol (a) (x + 17°) + 113° +

(2x − 33°) + 113° = (4 − 2) × 180°

Sol (a) 104° + ( ) +

( ) + ( ) =

[( ) − ( )] ×

180° (∠ sum of polygon)

3x + 210° = 360°

3x = 150°

x = 50°

( )

=

(b) ∠B = ∠D = 113°

∠A = x + 17°

= 50° + 17°

= 67°

∠C = 2x − 33°

= 2 × 50° − 33°

= 67°

∴ ∠A = ∠C = 67°

∴ ABCD is a

parallelogram.

given

from (a)

from (a)

opp. ∠s equal

(b) ∠P = ∠__ = ____

∠Q = 3a − 29°

=

given

12. In the figure, AC and BD intersect at E.

(a) Find y.

(b) Prove that ABCD is a parallelogram.

○○○○→→→→ Ex 7A 18, 19

A

B

C

D

113°

x + 17°

2x − 33°

113°

P

Q R

S

104°

2a + 6°

3a − 29°

104°

A

B

C

D 21

20 y

29

20 E

10

Level Up Questions

13. In the figure, AEB and DFC are straight lines. AEFD is

a parallelogram. Prove that ABCD is a parallelogram.

14. Refer to the figure.

(a) Prove that △BDA ≅ △DBC.

(b) Hence, prove that ABCD is a parallelogram.

Note the difference

between the

‘properties’ and

‘conditions’ of

parallelograms.

A

B C

D

E F

A

B

C

D

11

A B

C D

5 cm

8 cm

y cm

x cm

S

R

Q

P

10 cm 8x cm

5x cm y cm

A B

C D

3y cm 9 12 cx A

B C

D

2y + 15° x

65°

E F

G H

70° 80°

x

S

P

R

Q

3x 2x

y 125°

E

D

C B

A (2x + 3) cm

(4x − 1) cm 5 c(x + 3y) cm

x 2x

72°

9 cm

O

P

Q

R

S

3

ycm

New Century Mathematics (2nd Edition) 3B

7 Quadrilaterals

Level 1

Find the unknown(s) in each of the following parallelograms. [Nos. 1−−−−8]

1. 2.

3. 4.

AEC and BED are straight lines.

5. 6.

PTQ is a straight line.

7. 8.


POR and QOS are straight lines.


7A

12

9. In the figure, ABCD is a parallelogram. If CA = CD, find x.

10. In the figure, PQRS is a parallelogram. TPQ is a straight line. If

RQ = ST, find the perimeter of quadrilateral QRST.

11. In the figure, AGCB and CFED are parallelograms. BCD and

GFC are straight lines. Find x and y.

12. In the figure, △ABD ≅ △CDB. Prove that ABCD is a

parallelogram.

13. In the figure, PQRS and PQTU are parallelograms. Prove that

(a) SR = UT,

(b) RSUT is a parallelogram.

14. In the figure, ABCD is a parallelogram. AGC and BFGED are

straight lines. It is given that EG = FG. Prove that AFCE is a

parallelogram.

15. In the figure, PQRS is a parallelogram. PSU and TQR are

straight lines. It is given that SU = QT. Prove that PTRU is a

parallelogram.


(a) Find x.


40°

3 x c7 6

x + 40°

1196° − x

6 cm

8 cm

9 cm

13

Level 2

17. In the figure, ABCD is a parallelogram. CEB is a straight line.

It is given that AC = AD and AE ⊥ CB.

(a) Find ∠ADC.

(b) Find ∠BAE.

18. In the figure, ABCD is a parallelogram. FBA, FED and CEB

are straight lines. It is given that FB = EB.

(a) Find ∠CDF.

(b) Find ∠FAD.

19. In the figure, ABCD and AEDF are parallelograms. AEC and BED

are straight lines. Find x and y.

20. In the figure, PQRS is a parallelogram. If PR = 8 cm and the

perimeter of △PRS is 30 cm, find the perimeter of PQRS.

21. In the figure, ABCD is a parallelogram. AC and BD

intersect at E. CF ⊥ FB and BD ⊥ DA. If BE = 4 cm,

AD = 15 cm and FC = 9 cm, find the perimeter of pentagon

ABFCD.

22. In the figure, RTQU is a parallelogram. PTR and STQ are

straight lines. It is given that RU = ST and QU = PT. Prove that

PQRS is a parallelogram.

23. In the figure, ABCD is a parallelogram. DFC and CEB are

straight lines. It is given that ∠CEG = ∠CBA and FC = GE.

(a) Prove that CFGE is a parallelogram.

(b) If AD = 12 cm and BE = 4 cm, find FG.

50 50y cm 10 cm 25 35 8 cm

14


(a) Find the value of a.


(c) Find the value of b.

25. In the figure, ABCD is a parallelogram. AFGEC and BGD are

straight lines. It is given that ∠DFG = ∠BEG.

(a) Prove that △DGF ≅ △BGE.

(b) Prove that BEDF is a parallelogram.

26. In the figure, ABCD is a parallelogram. AED and BFC are

straight lines. What condition(s) about △ABF and △CDE

should be added so as to make AFCE a parallelogram? Explain

your answer.

a° + °

5a° + °

4a° − °

12 a + 9 3b − 1

15

Answer

Consolidation Exercise 7A

1. x = 8, y = 5

2. x = 2, y = 16

3. x = 9, y = 4

4. x = 65°, y = 50°

5. x = 30°

6. x = 25°, y = 50°

7. x = 2, y = 1

8. x = 24°, y = 27

9. 70°

10. 40 cm

11. x = 10, y = 120°

16. (a) 28°

17. (a) 65° (b) 25°

18. (a) 50° (b) 80°

19. x = 120°, y = 10

20. 44 cm

21. 70 cm

23. (b) 8 cm

24. (a) 20 (c) 10

26. △ABF ≅ △CDE

16

F3B: Chapter 7B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

17



(Full Solution)


7B Level 1


Teacher’s

Signature

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7B Level 2


Teacher’s

Signature

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( )


7B Level 3


Teacher’s

Signature

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( )


7B Multiple Choice


Signature

___________

( )


Self-Test


_________

18

Book 3B Lesson Worksheet 7B (Refer to §7.3)

7.3A Properties of Rhombuses

(a) Definition of a rhombus:

a quadrilateral with 4 equal sides

(b) Properties of rhombuses:

Property All properties of a

parallelogram.

Diagonals are

perpendicular to

each other.

Each interior angle is

bisected by a

diagonal.

Reference property of rhombus


In the figure, ABCD is a

rhombus. AEC and BED

are straight lines. Find the

unknowns.

In the figure, PQRS is a

rhombus. PTR and STQ

are straight lines. Find the

unknowns.

Sol ∵ AB = BC (by definition)

∴ x = 9

∵ AC ⊥ BD (property of rhombus)

∴ y = 90°

Sol m = ( of rhombus)

∵ RT = ____ ( of rhombus)

∴ n =

Find the unknowns in each of the following rhombuses. [Nos. 1−−−−2]

1.

PTR and STQ are straight lines.

2.


○○○○→→→→ Ex 7B 1, 6, 12, 13

P

Q

R

S

T m

4

5

n A

B

C

D

x y 52°

E

A rhombus has 4 equal

sides.

A rhombus is a

kind of

parallelogram.

A

B

C

D

9

x

y

E

P

Q R

S

T

m

56°

n

4

Including:

(i) Opposite sides

are equal.

(ii) Opposite angles

are equal.

(iii) Diagonals bisect

each other.

19

7.3B Properties of Rectangles

(a) Definition of a rectangle:

a quadrilateral with 4 equal interior angles

(b) Properties of rectangles:

Property

All properties of

a parallelogram.

All the interior

angles are right

angles.

Diagonals are equal. Diagonals bisect

each other into four

equal line segments.

AC = BD

Reference property of rectangle



rectangle. AEC and BED

are straight lines. Find

the unknowns.


rectangle. PTR and STQ


the unknowns.

Sol ∵ DE = EB (property of rectangle)

∴ x = 5

∵ AC = BD (property of rectangle)

∴ y = 5 + x

= 5 + 5

= 10

Sol ∠SRQ = _____ ( of rectangle)

( ) + m = ( )

m =

∵ ST = ____ ( of rectangle)

∴ n =

Find the unknowns in each of the following rectangles. [Nos. 3−−−−4]

3.


4.

EKG and HKF are straight lines.

○○○○→→→→ Ex 7B 2, 7, 10

A

B C

D

A

B C

D

5

x y

z

6 E

E

G

H

m 136°

n

F

K

A

B C

D

E y

x

5

P

Q R

S

m 35°

n 7

T

Which kind of

triangle is △BCD?

Which kind of

triangle is △GHK?

20

7.3C Properties of Squares

(a) Definition of a square:

a quadrilateral with four equal sides and four equal interior angles

(b) Properties of squares:

Property

All properties of a

rectangle.

All properties of a

rhombus.

Angle between a side

and a diagonal is 45°.

Reference property of square



square. Find the unknowns.


square. PTR and STQ


the unknowns.

Sol ∵ CD = BC (by definition)

∴ x = 10

y = 45° (property of square)

z = 90° (property of square)

Sol ∵ PS = ____ ( )

∴ 4p =

∵ SQ ⊥ ____ ( )

∴ n =

Find the unknowns in each of the following squares. [Nos. 5−−−−6]

5.

AED is a straight line.

6.

PTS, PWR, TWX and QXR are straight lines.

○○○○→→→→ Ex 7B 3, 8, 11

45°

A

B C

D

x

y

30°

E P

Q R

S

W

m

k

n

64°

T

X

A

B C

D

x

y

10

z

A square is a kind

of rectangle.

P

Q R

S

T n p + 9

4p

A square is a

kind of

∵ A square is a kind of

parallelogram. ∴ PS // ____

21

A D

C B

7.3D Properties of Other Special Quadrilaterals

I. Properties of Trapeziums

(a) Definition of a trapezium:

a quadrilateral with only one pair of parallel oppposite sides

(b) The figure shows an isosceles trapezium.

It has the following properties:

(i) ∠B = ∠C,

(ii) ∠A = ∠D,

(iii) AC = BD.

Find the unknowns in each of the following isosceles trapeziums. [Nos. 7−−−−8]

7.

∠P + ∠___ = 180° (_____ ∠s, PS // QR)

=

∵ PQRS is an isosceles trapezium. (given)

∴ ∠R = ∠__

k =

∠S = ∠__

=

8.


∵ ABCD is an isosceles trapezium. (given)

∴

○○○○→→→→ Ex 7B 4(a)

Q R

S P

78°

m n

k 2 5

4

y

E

A

B

D

C

x

22

II. Properties of Kites

(a) Definition of a kite:

a quadrilateral with two pairs of equal adjacent sides

(b) The figure shows a kite, where AC and BD intersect

at O. It has the following properties:

(i) ∠ABC = ∠ADC, ∠BAC = ∠DAC,

∠BCA = ∠DCA;

(ii) BO = DO;

(iii) AC ⊥ BD.

[Reference: property of kite]

In each of the following, ABCD is a kite. AC and BD intersect at O. Find m and n. [Nos. 9−−−−10] 9.

OB = ____ (property of kite)

m =

∠DAC = ∠______ (property of ____)

n =

10.

In each of the following, PQRS is a kite, where PQ = PS and RQ = RS. Find x and y. [Nos. 11−−−−12]

11. 12.

○○○○→→→→ Ex 7B 5

A

B

C

D O

m

21°

n

4

5

A

B

C

D m 41° n

103° O

P

Q

R

S

54°

x

110°

y

P

Q

R

S

x 48°

y

95°

A

B

C

D O

Recall:

Sum of the interior

angles of a polygon

= (n − 2) × _____

(∠ sum of polygon)

� In fact,

△ABC ≅ △ADC

(SSS).

23

�� ‘Explain Your Answer’ Question

13. In the figure, ABCD is a parallelogram. BCE is a straight line.

(a) Find ∠BCD.

(b) By definition, determine whether ABCD is a rectangle.

Explain your answer.

Level Up Question

14. Refer to the figure. Find DC.

(Hint: Draw a perpendicular line from D to BC.)

Trapezium ABCD has a

side perpendicular to

its two bases. It is

called a right-angled

trapezium.

A

B C

D

E

A

B C

D

12 cm

15 cm

50 cm

24

12 cm

x cm 3y cm

(z − 3) cm

5010 38

65 3y + 1

16 − 255 9 cm (z − 1) cm


7 Quadrilaterals

Level 1

1. Find the unknowns in each of the following rhombuses.

(a) (b)

(c) (d)

POR and SOQ are straight lines. EOG and HOF are straight lines.

2. Find the unknowns in each of the following rectangles.

(a) (b)

(c) (d)

POR and SOQ are straight lines.

POR and SOQ are straight lines.


7B

25

10 cm (b + 2) cm

a cm

22 3r c9 cm

6556 2x

29 11 42

5 x y cm 12

3. Find the unknowns in each of the following squares.

(a) (b)

(c) (d)

4. Find the unknowns in each of the following trapeziums.

(a) (b)

PTQ is a straight line.

5. Find the unknowns in each of the following kites.

(a) (b)

AEC and DEB are straight lines.

26

6. In the figure, ABCD is a rhombus. AC and BD intersect at E.

It is given that AC = 16 cm and BD = 30 cm. Find the

perimeter of rhombus ABCD.

7. In the figure, ABCD is a rectangle. AC and BD intersect at E.

It is given that AD = 16 and BE = 10. Find AB.

8. In the figure, PQRS is a square. PR and QS intersect at O. The

perimeter of PQRS is 36 cm.

(a) Find PQ.

(b) Find PO, correct to 3 significant figures.

9. In the figure, ABCD is a rectangle. E is a point on DB such that

EC = BC.

(a) Find ∠CEB.

(b) Find ∠ECD.

10. In the figure, ABCD is a square. DFB and AEF are straight lines. It

is given that AB = EB and ∠EAB = 75°.

(a) Find ∠AFB.

(b) Find ∠DBE.

11. In the figure, PQRS is a rhombus. STQ is a straight line. U is a

point on RQ such that TU ⊥ RQ. It is given that ∠SPQ = 110°.

(a) Find ∠PQS.

(b) Find ∠UTS.

16 cm 30 cm

1070

7511

27

12. The figure shows a kite ABCD, where AB = AD and CB = CD. It is given

that ∠ADC = 140° and ∠CAD = 20°.

(a) Find ∠ACD.

(b) By definition, determine whether ABCD is a rhombus. Explain your

answer.

13. In the figure, PQRS is a rectangle. It is given that PS = 6 cm and the

perimeter of PQRS is 24 cm.

(a) Find SR.

(b) By definition, determine whether PQRS is a square. Explain

your answer.

Level 2

14. In the figure, ABCD is a rhombus. The area of ABCD is 96 cm2 and

DB = 12 cm. Find the perimeter of ABCD.

15. In the figure, PQRS is a rhombus. T is a point on SR such that TP

is the angle bisector of ∠SPR. It is given that ∠PTS = 63°.

(a) Find ∠TPR.

(b) Find ∠PQR.

16. The figure shows a trapezium ABCD, where AB // DC and AD

= BC. DCE is a straight line. It is given that

∠DAC = 32° and ∠BCE = 66°. Find ∠ACD.

17. In the figure, ABCD and BGFD are rectangles. AEC, BED and

FCG are straight lines. It is given that ∠BAC = 24°.

(a) Find ∠BEC.

(b) Find ∠FDC.

18. In the figure, ABCD is a rectangle and CEDF is a rhombus.

AFC and BFD are straight lines. It is given that AD = 18 cm and

DE = 15 cm.

(a) Find DC.

(b) Find the perimeter of pentagon ABCED.

14 206 c12 cm

63

2418 cm 15 cm

32 66

28

19. In the figure, ABC, AFD and EFB are straight lines. ABDE is a

parallelogram and BCDE is a square. It is given that ED = 3 cm.

(a) Find ∠EAB.

(b) Find the perimeter of quadrilateral ACDE.

(c) Find FD.

(Give your answers correct to 3 significant figures if necessary.)

20. In the figure, ABDE is a square and BCD is an equilateral triangle.

EFC is a straight line.

(a) Find ∠DCE.

(b) Find ∠BFE.

21. In the figure, ABCD is a square. BFE and DFC are straight

lines. It is given that ∠FBC = 16°, ∠FDE = 77° and DE =

12 cm.

(a) Prove that △BDE is isosceles.

(b) Find the area of square ABCD.

22. In the figure, ABCD is a parallelogram and ADEF is a

rectangle.

(a) Is it true that ADEF must be a square? Explain your

answer.

(b) If BC = AF, find ∠AEF.

16 77 12 cm

3 cm

29

Answer

Consolidation Exercise 7B

1. (a) x = 12, y = 4, z = 15

(b) x = 50°, y = 50°, z = 80°

(c) x = 6, y = 8, z = 10

(d) x = 90°, y = 52°, z = 52°

2. (a) x = 90°, y = 25°

(b) x = 2, y = 3

(c) x = 55°, y = 110°, z = 35°

(d) x = 3, y = 12, z = 10

3. (a) a = 10, b = 8

(b) x = 90°, y = 90°, z = 45°

(c) p = 23°, q = 112°, r = 3

(d) p = 25°, q = 45°, r = 70°

4. (a) x = 62°, y = 124°

(b) x = 5, y = 7

5. (a) x = 9, y = 90°, z = 29°

(b) x = 21°, y = 28°, z = 26°

6. 68 cm

7. 12

8. (a) 9 cm (b) 6.36 cm

9. (a) 70° (b) 50°

10. (a) 60° (b) 15°

11. (a) 35° (b) 125°

12. (a) 20° (b) yes

13. (a) 6 cm (b) yes

14. 40 cm

15. (a) 21° (b) 96°

16. 34°

17. (a) 48° (b) 66°

18. (a) 24 cm (b) 90 cm

19. (a) 45° (b) 16.2 cm (c) 3.35 cm

20. (a) 15° (b) 105°

21. (b) 72 cm2

22. (a) no (b) 45°

30

F3B: Chapter 7C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 16


(Video Teaching)

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Book Example 18


(Video Teaching)



(Full Solution)


7C Level 1


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31

Book 3B Lesson Worksheet 7C (Refer to §7.4)

7.4 Proofs Related to Parallelograms

Example 1 Instant Drill 1 In the figure, ABCD is a parallelogram. Prove

that △ADE ≅ △CBF.

In the figure, ABCD is a parallelogram. Prove

that △AED ≅ △CFB.

Sol In △ADE and △CBF,

∠ADE = ∠CBF

AD = CB

∠DAE = ∠BCF

∴ △ADE ≅ △CBF

given

opp. sides of //gram

opp. ∠s of //gram

ASA

Sol In △AED and △CFB,

AE = ____

∠DAE = ∠____

given

_________ of //gram

1. In the figure. ABCD is a parallelogram.

AC, BD and EG intersect at F.

(a) Prove that △AEF ≅ △CGF.

(b) Prove that AE = CG.

2. In the figure. ABCD is a parallelogram.

AEFC is a straight line.

(a) Prove that △ADF ≅ △CBE.

(b) Prove that BE // FD.

A B

C D

E

F A

B C

D

E F

A

B C

D

E

F G

A

B C

D

E

F

32

3. In the figure, ABCF is a parallelogram.

Prove that ED // BC.

4. In the figure, ABCD is a rhombus and

AEFC is a straight line. Prove that

△ABE ≅ △CDF.

○○○○→→→→ Ex 7C 1–6

5. In the figure, ABCD is a rectangle.

(a) Prove that △ABE ≅ △DCE.

(b) Prove that △BEC is an isosceles

triangle.

6. In the figure, ABCD is a square.

(a) Prove that △ADE ≅ △CDF.

(b) Prove that ∠ADF = ∠CDE.

○○○○→→→→ Ex 7C 7–10

A

B C

D E

F

A

B

C

E

D

F

A

B

E D

C

A

B

E

D

F C

33

Level Up Questions

7. In the figure, ABCD is a parallelogram with a diagonal BEFD.

(a) Prove that △DAF ≅ △BCE.

(b) If BF = 5 cm and EF = 1 cm, find BD.

8. In the figure, ABCD is a square.

(a) Prove that △EBF ≅ △ECF.

(b) Prove that △EAD is an isosceles triangle.

A

B C

D

E

F

A

B C

E

D

F

34


7 Quadrilaterals

Level 1

1. In the figure, ADE, GFE, DFC and AGB are straight lines. ABCD

is a parallelogram and ∠DFE = ∠BCD. Prove that ED = EF.

2. In the figure, ABCD is a parallelogram. AEB and DFC are

straight lines. It is given that ∠ADE = ∠CBF. Prove that

ED // BF.

3. In the figure, AGD and EDC are straight lines. ABCD and

DEFG are both parallelograms. Prove that ∠DAB = ∠EFG.

4. In the figure, ABC is a straight line. ABDE is a rectangle and

BCDE is a parallelogram. Prove that AB = BC.

5. In the figure, ABCD is a parallelogram. M and N are the

mid-points of AD and BC respectively. Prove that

△ABN ≅ △CDM.


7C

35

6. In the figure, ABCD is a rhombus. E is a point lying on AC. Prove

that △ABE ≅ △ADE.

7. In the figure, ABC and FED are straight lines. ABEF and BCDE

are both squares.

(a) Prove that △BED ≅ △BEF.

(b) Prove that △FBD is a right-angled isosceles triangle.

8. In the figure, BPCS is a straight line. ABCD and PQRS are two

identical rectangles.

(a) Prove that △BPQ ≅ △SCD.

(b) Prove that BQ // DS.

Level 2

9. In the figure, ABCD is a rectangle. E is the mid-point of AD. AC

and BE intersect at F.

(a) Prove that △AFE ~ △CFB.

(b) Prove that FC = 2AF.

10. In the figure, ABCD and BFDE are both parallelograms. AC

intersects DE and FB at G and H respectively.

(a) Prove that △AGE ≅ △CHF.

(b) Prove that △AGD ≅ △CHB.

11. In the figure, ABCD is a square. AC and BD intersect at F. It is given

that △AED ≅ △AFD. Prove that AFDE is a square.

36

12. In the figure, ABCD is a rectangle. E and F are points on AB and DC

respectively. AC and EF intersect at G. It is given that AE = FC and

EB2 + BC2 = AE2. Prove that EF ⊥ AC.

13. In the figure, ABCD is a square. E and F are points on AB and BC

respectively. CE and DF intersect at G. It is given that AE = BF.

(a) Prove that △EBC ≅ △FCD.

(b) Prove that EC ⊥ FD.

14. In the figure, ABCD is a rhombus. QBDP is a straight line

and ∠QAB = ∠PCD.

(a) (i) Prove that AQ = CP.

(ii) Prove that AQ // CP.

(b) Prove that AQCP is a rhombus.

37

F3B: Chapter 7D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 19


(Video Teaching)

Book Example 20


(Video Teaching)

Book Example 21


(Video Teaching)

Book Example 22


(Video Teaching)

Book Example 23


(Video Teaching)

Book Example 24


(Video Teaching)



(Full Solution)


7D Level 1


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38


7D Level 3


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7D Multiple Choice


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Self-Test


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39

Book 3B Lesson Worksheet 7D (Refer to §7.5)

7.5A Mid-point Theorem

In △ABC,

if AM = MB and AN = NC, then

(a) MN // BC,

(b) MN =

2

1BC.

[Reference: mid-pt. theorem]

Example 1 Instant Drill 1 In the figure, M and N are

the mid-points of AB and

AC respectively. Find the

unknowns.

In the figure, M and N are

the mid-points of BC and

AB respectively. Find the

unknowns.

Sol ∵ AM = MB

and AN = NC. (given)

∴ MN // BC (mid-pt. theorem)

x = ∠ACB (corr. ∠s, MN // BC)

= 44°

MN =

2

1BC (mid-pt. theorem)

y =

2

1× 10

= 5

Sol ∵ AN = ____

and BM = ____. (given)

∴ ____ // ____ (__________ theorem)

∴ x = ∠BMN (_________________)

=

MN =

2

1×____ (mid-pt. theorem)

=

Find the unknown(s) in each of the following figures. [Nos. 1−−−−2]

1.

ANC and AMB are straight lines.

2.

APB, AQC and CRD are straight lines.

○○○○→→→→ Ex 7D 1–6

A

B

N 60°

x 4 cm

C

M

4 cm

3 cm

3 cm

74°

A

B

P

C

Q

D

R

x cm

y cm

11 cm

4.5 cm

A

B

M N

C

A

B

M N

C 44°

x

y cm

10 cm B C

N

85° x

y cm

3 cm

A

M

Consider △ABC

and △ACD one

by one.

40

7.5B Intercept Theorem

In the figure,

if AB // CD // EF and AC = CE,

then BD = DF.

[Reference: intercept theorem]


In the figure, ACE and

BDF are straight lines.

Find x.

In the figure, PRT and

QSU are straight lines.

Find y.

Sol ∵ AB // CD // EF

and BD = DF = 8 cm. (given)

∴ AC = CE (intercept theorem)

x = 7

Sol ∵ PQ // ____ // ____

and QS = ____ = ( ) cm. (given)

∴ RT = ____ ( )

y =


3.

PRT and QSU are straight lines.

4.

ABCD and EFGH are straight lines.

○○○○→→→→ Ex 7D 7–9

P Q

T

R S

U

5 cm

y cm

A

B

C

E

D H

G

F

2 cm

x cm

4 cm

3 cm

3 cm

y cm

A B

E

C D

F

L1 L2

A B

E

C D

F

8 cm

8 cm 7 cm

x cm P Q

T

R S 11 cm 11 cm

9 cm y cm U

41

In △ABC,

if AM = MB and MN // BC,

then AN = NC.

[Reference: intercept theorem]


In the figure, ABC and AED are straight lines.

Find ED.

In the figure, ABC and CDE are straight lines.

Find BC.

Sol ∵ AB = BC = 4 cm

and BE // CD. (given)

∴ ED = AE (intercept theorem)

= 2.5 cm

Sol ∵ CD = ____ = ( ) cm

and BD // ____. ( )

∴ BC = ____ ( )

=


5.

ADB and AEC are straight lines.

6.

ABCD and AEFG are straight lines.

A

B

C E

D

10 cm

x cm

A

B E

C

D

F

G

7 cm 6 cm

6 cm

6 cm

x cm

y cm

A

B

M N

C

A

B

C

E

D

4 cm

4 cm

2.5 cm 4.5 cm

7 cm

7 cm

A

B

C

E

D

42

7.

PQR, PTS and RTU are straight lines.

8.

ADB and CEB are straight lines.

○○○○→→→→ Ex 7D 10–13

9. In the figure, ABC and AED are straight lines.

(a) Find AE. (b) Find CD.

○○○○→→→→ Ex 7D 14


10. In the figure, X, Y and Z are the mid-points of PQ, PR and PS

respectively. A student claims that if XY = YZ, then QR = RS. Do

you agree? Explain your answer.

6 cm 10 cm

6 cm

x cm

y cm

P

Q

R S

T

U

5 cm

A

B

E

5 cm

C

D x cm

A

B

C

E

D

9 cm 12 cm

P

Q R

S X

Y

Z

Consider △PRS

and △PRU one

Do not mix up the mid-point theorem with

the intercept theorem.

Mid-point theorem: Intercept theorem:

(a) BE // CD AE = ED

(b) BE =2

1CD

Is AC parallel

to DE?

A

C D

E B

A

C D

B E

43

Level Up Questions

11. In the figure, ABC, AED and CEF are straight lines.

(a) Prove that AB = BC.

(b) Find BE.

12. In the figure, ABCD and AGFE are straight lines.

(a) Find CF.

(b) Find the perimeter of quadrilateral BCFG.

A

B

C

E

D

F

30 cm

Is AF parallel to

BE?

7 cm A

B

C

E D

F

G 7 cm

14 cm 18 cm

10 cm

44

5 cm

5 cm

8 cm 8 cm

9 cm x cm

12 cm y c

65 6572x cm 15 cm

13 85


7 Quadrilaterals

Level 1


1. 2.

APB and AQC are straight lines. AMB and ANC are straight lines.

3. 4.

CDA and CEB are straight lines. ADB and AEC are straight lines. 5.

APB and AQC are straight lines.


7D

45

x c4 c x c2 c2 c2 c 3 c y c

x cm 5 cm 2x cm

6 cm 6 cm y cm

x cm y cm

9 cm 8 cm

11 cm 11 cm

y cm 5 cm


6. 7.

ACE and BDF are straight lines. ABCD and EFGH are straight lines.

8. 9.

ABCD and AGFE are straight lines. ACE, AKF and BDF are straight lines.

10.

ACF, BDF and BEG are straight lines.

11. In the figure, ACE, ADF and BDE are straight lines. It is given

that AB // CD // EF and AD = DF. Find x and y.

6 cm x cm

46

12. In the figure, AEB, AFC, AGD, EFG and BCD are straight

lines. It is given that AF = FC.

(a) Prove that EG // BD.

(b) Find GD and FG.

Level 2

13. In the figure, AGC, DGE and BECF are straight lines. It

is given that BA // ED and CA // FD.

(a) Find GE.

(b) Prove that BE = EC = CF.

14. In the figure, G is the mid-point of AC, while F is the

mid-point of both BC and AE. ABD is a straight line.

(a) Prove that BC // DE.

(b) Find ∠EAD.

15. In the figure, ACE and BDF are straight lines. It is given that

AB // CD // EF and AC = CE. Find DF and CD.

(Hint: Join AF or BE.)

16. In the figure, E is the mid-point of AB and AD // EF // BC. AC

and BD intersect at F.

(a) Prove that AF = FC and BF = FD.


5 cm 5 cm 3 cm

8 cm

3 cm 4 cm 4 cm

6 cm

62 824 c8 c

22 cm

16 cm

6 c

47

17. In the figure, D, E and F are the mid-points of AB, BC and AC

respectively.

(a) Prove that △BED ≅ △FDE.

(b) Prove that ∠DBE = ∠EFD.

18. In the figure, D and E are the mid-points of AB and AC

respectively. DC and BE intersect at F.

(a) Prove that △FDE ~ △FCB.

(b) Is F the mid-point of BE? Explain your answer.

19. In the figure, AF and BE intersect CD at H and G

respectively. ACE and BDF are straight lines. It is given that

AB // CD // EF, AC = CE, EF = 26 cm and GH = 8 cm.

(a) Find AB.

(b) Find CD.

20. In the figure, ADBG, AEC, EFG and BFC are straight lines. It is

given that DE // BC and AD = DB = BG. Prove that BF =

3

1FC.

21. In the figure, ADB, AEGC, BCF and DGF are straight lines. It is

given that AD = DB, DE // BF and BC : CF = 2 : 1.

(a) Prove that △FCG ≅ △DEG.

(b) Find AG : GC.

8 c 26 cm

48

Answer

Consolidation Exercise 7D

1. 6

2. 18

3. 65°

4. 43°

5. x = 7.5, y = 48°

6. 4

7. x = 3, y = 3

8. x = 5, y = 12

9. x = 8, y = 9

10. x = 4.5, y = 5

11. x = 6, y = 8

12. (b) GD = 8 cm, FG = 1.5 cm

13. (a) 3 cm

14. (b) 36°

15. DF = 6 cm, CD = 19 cm

18. (b) no

19. (a) 10 cm (b) 18 cm

21. (b) 3 : 1

49

F3B: Chapter 8A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


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Book Example 2


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(Full Solution)


8A Level 1


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8A Level 3


Teacher’s

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8A Multiple Choice


Signature

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Self-Test


_________

50


[In this worksheet, the contents of Questions 1, 5–7 are beyond the scope of the curriculum.]

8.1A Reflectional Symmetry of 3-D Figures

If a solid can be divided by a plane into two identical parts and these two parts are

the mirror images of each other, the solid is said to have reflectional symmetry.

The plane is called the plane of reflection.

e.g.

1. In each of the following, determine whether the shaded plane is a plane of reflection of the

given 3-D figure.

(a)

(yes / no)

(b)

(yes / no)

(c)

(yes / no)

(d)

(yes / no)

(e)

(yes / no)

(f)

(yes / no)

� A solid may

have more than

one plane of

reflection.

51


Determine whether each of the following is a

plane of reflection of the cube ABCDEFGH.

(a) the plane containing EF and HG

(b) the plane passing through the mid-points

of AB, DC, EH and FG

Sol

(a) It is not a plane of reflection of the

cube.

(b) It is a plane of reflection of the cube.

Determine whether each of the following is a

plane of reflection of the regular tetrahedron

ABCD.

(a) the plane containing AB and passing

through the mid-point of CD

(b) the plane passing through the mid-points

of AB, BC and CD

Sol

(a) It (is / is not) a plane of reflection of

the regular tetrahedron.

(b) It (is / is not) a plane of reflection of

the regular tetrahedron.

2. Refer to the cube in Example 1. Determine whether each of the following is a plane of

reflection of the cube.

(a) the plane containing AF and BG (yes / no)

(b) the plane containing AB and EH (yes / no)

(c) the plane passing through the mid-points of AD, BC, FE and GH (yes / no)

3. Refer to the regular tetrahedron in Instant Drill 1. Determine whether each of the following is

a plane of reflection of the regular tetrahedron.

(a) the plane containing BC and passing through the mid-point of AD (yes / no)

(b) the plane passing through A and the mid-points of BC and CD (yes / no)

○○○○→→→→ Ex 8A 7, 8

A D

E F

G

B C

H

(a) (b)

D

E F

G

B C

H

A

E F

G H

D

B C

A

A

D B

C

(a) (b) A

D B

C

A

D B

C

� A cube has 6

identical square

faces.

� A regular

tetrahedron has

4 identical faces

of equilateral

triangles.

Indicate the plane

on the cube

first.

52

4. Draw all the planes of reflection of a cube in the following.

(Two of the planes of reflection have been drawn for you as examples.)

How many planes of reflection does a cube have?

5. Does each of the following 3-D figures have reflectional symmetry? If yes, write down the

number of planes of reflection. (a) (b) (c)

○○○○→→→→ Ex 8A 1–3

8.1B Rotational Symmetry of 3-D Figures

If a solid coincides with its original figure n times (n > 1) in one complete

revolution (i.e. 360°) about a straight line, the solid is said to have rotational

symmetry of order n. The line is called the axis of rotation.

e.g. Consider the right square prism below.

Note: An object can have different axes of rotation,

and the corresponding orders of rotational

symmetry can also be different.

axis of rotation

(n = 2)

P 1ℓ Consider the top face.

rotate

1

8

rotate

18

0°

1st coincidence 2nd coincidence

P

P

P

axis of rotation

(n = 4)

2ℓ

53

6. In each of the following, determine whether the straight line ℓ is an axis of rotation of the given

3-D figure. If yes, write down the corresponding order of rotational symmetry about ℓ .

(a)

(b)

� yes, order of rotational symmetry =

� no


� no

(c)

(d)


� no


� no

○○○○→→→→ Ex 8A 4–6

Level Up Questions

7. The given figure shows a right triangular prism.

(a) Does it have reflectional symmetry?

(b) Does it have rotational symmetry?

8. In each of the following 3-D figures, X and Y are the mid-points of two edges, and Z is the

centre of the shaded face. Determine whether lines XY and AZ are the axes of rotation of the

given 3-D figures. If yes, write down the corresponding order of rotational symmetry.

(a) Cube ABCDEFGH (b) Regular tetrahedron ABCD

ℓ ℓ

ℓ

ℓ

A B

G F

E

D C

H

X

Z

Y

D

A B

C

Z

X

Y

54

P

Q

R

S

N


8 More about 3-D Figures

Level 1

In each of the following prisms, write down the number of planes of reflection. [Nos. 1−−−−2]

1. 2.

In each of the following 3-D figures, ℓ is the axis of rotation. Find the orders of rotational symmetry

of the figures about ℓ . [Nos. 3−−−−4]

3. 4.

5. In each of the following, determine whether the shaded plane is a plane of reflection of the cube

ABCDEFGH.

(a) (b) (c)

6. The figure shows a regular tetrahedron PQRS. N is the mid-point

of QR. Determine whether each of the following is a plane of

reflection of the regular tetrahedron.

(a) plane PQS

(b) plane PSN

ℓ

ℓ

A B

C D

E

F G

H

A

C D

E

F G

H

B A B

D

E

F G

H

C


8A

55

A B

C D

E

F G

H

P

R

Q

P

Q

R

S

U

Y Z

X

Level 2

7. The figure shows a cube ABCDEFGH. P, Q and R are the

mid-points of the edges AB, BC and EH respectively.

Determine whether each of the following is an axis of rotation

of the cube. If yes, write down its order of rotational symmetry.

(a) line PR

(b) line QR

(c) line AH

8. The figure shows a regular tetrahedron PQRS. X, Y and Z are

the mid-points of the edges PS, SR and QR respectively. U is

the centre of the face PQR.

Determine whether each of the following is an axis of rotation

of the tetrahedron. If yes, write down its order of rotational

symmetry.

(a) line SU

(b) line XY

(c) line XZ

(d) line UY

9. The figure shows a cube with one face in grey and five faces in

white. If we take the colours into account,

(a) how many plane(s) of reflection does the cube have?

(b) how many axis/axes of rotation does the cube have? Find

the corresponding order of rotational symmetry.

56

Answer


1. 1

2. 2

3. 2

4. 3

5. (a) no (b) yes (c) no

6. (a) no (b) yes

7. (a) yes, 2 (b) no (c) yes, 3

8. (a) yes, 3 (b) no

(c) yes, 2 (d) no

9. (a) 4

(b) number of axis = 1, order = 4

57

F3B: Chapter 8B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)



(Full Solution)


8B Level 1


Teacher’s

Signature

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( )


8B Level 2


Teacher’s

Signature

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8B Level 3


Teacher’s

Signature

___________

( )


8B Multiple Choice


Signature

___________

( )


Self-Test


_________

58


8.2 Nets of 3-D Figures

(a) A 3-D figure can be folded from different nets.

(b) To determine whether a given figure can be folded into a specified 3-D figure, we

have to consider the relative positions of the points, lines and faces.

1. Which of the following figures can be folded into the polyhedron on the right?

○○○○→→→→ Ex 8B 1, 2


The following shows a prism and its net.

Sketch another net of the prism.

Sol Another net is shown below:

(or other reasonable answers)

The following shows a tetrahedron and its net.

Sketch another net of the tetrahedron.

Sol

2. A cube is shown on the right. Sketch two different nets of the cube.

○○○○→→→→ Ex 8B 5, 6

A B C

59


If we fold the net above into a polyhedron,

(a) which point will coincide with point A?

(b) which line segment will be stuck together

with BC?

Sol

(a) Point G will coincide with point A.

(b) Line segment DC will be stuck

together with BC.

If we fold the net above into a polyhedron,

(a) which point will coincide with point A?


with FE?

Sol

(a) Point will coincide with point A.

(b) Line segment will be stuck

together with FE.

3. If we fold the net on the right into a polyhedron,

(a) which point(s) will coincide with point F ?

(b) which line segment will be stuck together with DE?

4. If we fold the net on the right into a polyhedron,

(a) which point(s) will coincide with point C?


with HI?

○○○○→→→→ Ex 8B 7, 8

H

B

D

A

C

F

G

E

A

G

C

D

B

H

E

F

A, G

H

B, D, F E

C

A

C

J B

E

D

I

F G

H

?

A B

C

D

E

J

I

H

G F

A

C

E B

K

N

D

J M

L

G

H

F

I

B

A

C

F

D

E

G

H

I

J

60

5. The net on the right is folded into a dice.

(a) Which number is on the face opposite to the face with the

number ‘3’?

(b) Which numbers are on the four faces adjacent to the face with

the number ‘1’?

○○○○→→→→ Ex 8B 9, 10

Level Up Questions

6. Sketch a net of the polyhedron on the right.

7. The net on the right is folded into a dice.

(a) Sketch the symbol on the face opposite to the face with ‘ ’.

(b) Sketch the symbols on the faces that will meet at ‘ ’.

1

2

3

4 5

6

61



Level 1

1.

Which of the following figures can be folded into the cuboid above?

2.

Which of the following figures can be folded into the polyhedron above?

3. Which of the following rectangular blocks can be folded from the

net shown on the right?

P Q R

P Q R

P Q R


8B

62

4. The following figure shows a 3-D figure with a triangular base and its net. Sketch another net of

the 3-D figure.

5. A square prism and its net are shown below. Sketch another net of the prism.

6. If we fold the net below into a polyhedron, which three points will coincide with one another?

7.

If we fold the net above into a polyhedron, which line segment will be stuck together with

(a) GH?

(b) DE?

B A

C

D

E

F

B

A

C

E F

D

H G

I

J

63

AAAA

BBBB CCCC DDDD

EEEE FFFF

37.5 cm 20 cm 15 cm

42.5 cm 30 cm Fig. A Fig. B

8.

The net above is folded into a dice. Which letter is on the face opposite to the face with the

letter ‘C’?

Level 2

9. Kelvin wants to make a paper file holder with an opening as shown in Fig. A. Label the

appropriate lengths for Kelvin’s net in Fig. B.

10. Sketch a net of each of the following polyhedra.

(a) (b)

11. Sketch a net of each of the following polyhedra.

(a) (b)

64

12. The following net is folded into a dice.

(a) What is the pattern on the face opposite to the face with ‘ ’?

(b) What are the patterns on the four faces adjacent to the face with ‘ ’?

13. The net below is folded into a dice.

Find the numbers on the faces that will meet at

(a) the point X,

(b) the point Y.

14. The net below is folded into a polyhedron.

(a) Which point(s) will coincide with C?

(b) Which line segment will be stuck together with GF?

1111

2222

3333

4444

5555

6666

C

A

E D

B H G

F

65

Answer


1. P

2. Q

3. Q

6. B, D, F

7. (a) IH (b) BA

8. F

12. (a) ‘ ’

(b) ‘ ’, ‘ ’, ‘ ’, ‘ ’

13. (a) 3, 4, 5 (b) 1, 2, 3

14. (a) A, G (b) CD

66

F3B: Chapter 8C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)


8C Level 1


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8C Level 2


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8C Level 3


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8C Multiple Choice


Signature

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Self-Test


_________

67


8.3A Front, Top and Side Views of 3-D Objects

We can observe a 3-D object from its front, its top or its side.

e.g.

Front view: Top view: Side view:


The 3-D object below is made up of four

identical cubes. Draw the front, top and side

views of the object.

Sol Front view: Top view:

Side view:

The 3-D object below is made up of four

identical cubes. Draw the front, top and side


Sol Front view: Top view:

Side view:

front

top

side

front

top

side front

top

side

68

The following 3-D objects are made up of five identical cubes. Draw the front, top and side views

of each of the objects. [Nos. 1–2] 1. Front view: Top view: Side view:

2. Front view: Top view: Side view:

○○○○→→→→ Ex 8C 1, 3–5

3. The 3-D object below is made up of four identical cubes and a triangular prism. Draw the front,

top and side views of the object.


○○○○→→→→ Ex 8C 2, 6, 7

4. The 3-D object below is made up of six identical cubes and a half-cylinder. Draw the front, top

and side views of the object.


○○○○→→→→ Ex 8C 8

front

top

side

top

front

side

front

top

side

top

front

side

69

8.3B Identifying 3-D Objects from Given 2-D Representations

To identify the shape of a 3-D object, we need to know its 2-D representations

from three different views: the front, the top and the side.


Sketch a 3-D object that can satisfy the

following front, top and side views.

front top side

Sol

The required 3-D object is shown below:



front top side

Sol


Step 1111: by front view,

front Step 2222: by top view,

top


front Step 2222: by top view,

top

side

Step 3333: by side view,

side


front

top

side

70

5. Sketch a 3-D object that can satisfy the following front,

top and side views.

front top side




front top side

Sol




front top side

Sol



front

side


Step 2222: by top view,

top

front

top

side


front

side


Step 2222: by top view,

top

71

6. Sketch a 3-D object that can satisfy the following front,

top and side views.

front top side

○○○○→→→→ Ex 8C 9–11


Sketch a 3-D object that can satisfy the given

front, top and side views of the object.

front top side

Sol The required 3-D object is shown

below:

Sketch a 3-D object that can satisfy the given

front, top and side views of the object.

front top side

Sol The required 3-D object is shown below:

In each of the following, sketch a 3-D object that can satisfy the given front, top and side views of

the object. [Nos. 7–8]

7.

front top side

8.

front top side

○○○○→→→→ Ex 8C 12–15

front

top

side

72

Level Up Questions

9. The 3-D object on the right is made up of ten identical cubes.

Draw the front, top and side views of the object.


10. A 3-D object is made up of some identical cubes. The figures below show three of its 2-D

representations.

front top side

(a) Sketch a 3-D object that can satisfy the given information.

(b) How many cubes are there in the object?

front axis of

top

side

73



Level 1

In Nos. 1−−−−2, draw the front, top and side views of the given 3-D objects.

1. 2.

The following 3-D objects are made up of identical cubes. Draw the front, top and side views of

each of the objects. [Nos. 3–4]

3. 4.

5. The following 3-D object is made up of four identical cubes and a triangular prism. Draw the

front, top and side views.

top

front

side

top

side

front

top

side

front

top

side

front

top

side front


8C

74

In each of Nos. 6–8, sketch a 3-D object that can satisfy each of the following front, top and side


Front Top Side

6.

7.

8.

In each of Nos. 9–10, sketch a 3-D object that can satisfy the given front, top and side views of the

object.

Front Top Side

9.

10.

Level 2

In Nos. 11−−−−14, draw the front, top and side views of the given 3-D objects.

11. 12. top

side front

top

side

front

75

13. 14.

In each of Nos. 15–16, sketch a 3-D object that can satisfy the given front, top and side views of

the object.

Front Top Side

15.

16.

17. A 3-D object is made up of identical cubes. The figures below show three of its 2-D

representations.



18. A 3-D object is made up of identical cubes. The figures below show three of its 2-D

representations.



top

ffff sssstop

side

front

front

top

side

front

top

side

76

Answer

Consolidation Exercise 8C

17. (b) 4

18. (b) 7

77

F3B: Chapter 8D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)



(Full Solution)


8D Level 1


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8D Level 2


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8D Level 3


Teacher’s

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8D Multiple Choice


Signature

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Self-Test


_________

78


8.4A Distance between a Point and a Straight Line


PP′ is the distance between point P and the straight line ℓ .

1. The figure shows a cuboid. Find the distance between

(a) point D and line EF,

(b) point F and line HG,

(c) point H and line FG.

2. The figure shows a cuboid. Name the line segment which represents

the distance between

(a) point A and line DE,

(b) point F and line GB.

○○○○→→→→ Ex 8D 2

8.4B Relationship between Two Straight Lines

(a) In the figure, 1ℓ // 2ℓ . We say that d is the distance

between 1ℓ and 2ℓ .

(b) In the figure, 1ℓ and 2ℓ are two non-parallel straight

lines which lie in the same plane.

(i) P is the point of intersection of 1ℓ and 2ℓ .

(ii) θ (θ ≤ 90°) is the angle between 1ℓ and 2ℓ .

G

B

C

H

F

E

D

A

D

H

C

A B

G F

E 8 cm

3 cm

6 cm

3 cm F

D

E

A___ ⊥ DE

79

3. The figure shows a cuboid.

(a) Name the line segment which represents the distance between

the lines

(i) PQ and SR,

(ii) ST and RW.

(b) Name the angle between the lines

(i) PQ and PR,

(ii) PV and VQ.

4. The figure shows a right prism. Name the angle between the lines

(a) EC and DC,

(b) EB and BC,

(c) EB and AB.

○○○○→→→→ Ex 8D 3, 4

8.4C Relationship between a Point and a Plane

In the figure, P is a point outside plane π and

P′ is a point on plane π.

If PP′ is perpendicular to every line on π

which passes through P′, then:

(i) P′ is the projection of P on the plane π,

(ii) the length of PP′ is the distance between

P and the plane π.

5. The figure shows a right prism.

(a) Find the projection of

(i) point B on the plane DEF,

(ii) point C on the plane ABFE.

(b) Name the line segment which

represents the distance between

(i) point B and the plane DEF,

(ii) point C and the plane ABFE.

� PP′ ⊥ 1ℓ , PP′ ⊥ 2ℓ .

Q P

R S

Q P

U V

W

R S

T

Q P

R S

E

C B

F

A D

B

D

E F

E C

B

F

A

E

C

B

F

A

D

80

6. The figure shows a cube. Find the projection of

(a) point B on the plane ADEF,

(b) point E on the plane ABGF.

7. The figure shows a right prism with a trapezoidal base.

Find the projection of point A on the plane

(a) EFGH,

(b) CDEH.

○○○○→→→→ Ex 8D 5, 7(a)

8.4D Relationship between a Straight Line and a Plane

In the figure, straight line ℓ and plane π

intersect at point O. P is a point on the line ℓ

and Q is the projection of P on the plane π.

Then:

(i) OQ is the projection of OP on the plane π,

(ii) θ is the angle between the line ℓ and the

plane π.

e.g. The figure shows a cuboid ABCDEFGH.

(i) DE is the projection of DF on the

plane CDEH.

(ii) ∠FDE is the angle between line DF and

the plane CDEH.

G F

A B

C

H E

D M

G F

A B

C

H E

D

G B

C H

F

E D

A

projection

of OP on π

81


The figure shows a cube.

(a) Find the projection of CF on the plane

EFGH.

(b) Name the angle between line CF and the

plane EFGH.

Sol

(a) CF and the plane EFGH intersect at

F, and the projection of C on the plane

EFGH is H.

∴ HF is the projection of CF

on the plane EFGH.

(b) ∠CFH is the angle between line CF

and the plane EFGH.

The figure shows a right prism.

(a) Find the projection of PU on the plane

QRSU.

(b) Name the angle between line PU and the

plane QRSU.

Sol

(a) PU and the plane QRSU intersect

at , and the projection of P on

the plane QRSU is .

∴ is the projection of PU

on the plane QRSU.

(b) is the angle between line PU

and the plane QRSU.

8. Refer to the cube in Example 1.

(a) Find the projection of CF on the plane

CDEH.

(b) Name the angle between line CF and

the plane CDEH.

○○○○→→→→ Ex 8D 6, 7(b)

9. Refer to the prism in Instant Drill 1.

Name the angle between PU and each of

the following planes.

(a) PRST (b) PQR

○○○○→→→→ Ex 8D 8, 9

A D

E F

G

B C

H

P

Q R

S

T

U

CF intersects

EFGH at

F.

H is the

projection

of C on

EFGH.

HF is the

projection

of CF on

EFGH.

E F

G

C

H

E F

G

C

H

E F

G

C

H

P

Q

R

S U

P

Q

R

S U

P

Q R

S U

PU intersects

QRSU at

___.

___ is the

projection

of P on

QRSU.

___ is the

projection

of PU on

QRSU.

82

8.4E Relationship between Two Planes

In the figure, α and β are two non-parallel planes which

intersect at a straight line AB (line of intersection). Since

PQ ⊥ AB and PR ⊥ AB, we say that

∠RPQ is the angle between the planes α and β.

10. A cuboid is shown on the right. AC meets BD at X while

EG meets FH at Y. Name the lines of intersection for the

following pairs of planes.

(a) BADC and BGHC

(b) BGFA and AFHC

(c) BGED and AFHC

○○○○→→→→ Ex 8D 10


The figure shows a cuboid. Name the angle

between the planes ABHE and EFGH.

Sol

EH is the line of intersection of the planes

ABHE and EFGH.

∵ AE ⊥ EH and FE ⊥ EH.

∴ ∠AEF is the angle between the planes

ABHE and EFGH.

The figure shows a cube. Name the angle

between the planes VQST and WRST.

Sol

is the line of intersection of the

planes VQST and WRST.

∵ ⊥ and ⊥ .

∴ is the angle between the

planes VQST and WRST.

B

A

C

D

G

F

H

E

P U

T S

R

Q V

W

G

A

B

C H

D

G

F A

B

C H

E D

X Y

EH is the line of

intersection of

ABHE and EFGH.

∵ AE ⊥ EH and

FE ⊥ EH. ∴ ∠AEF is the angle

between the planes.

B

A H

F

G

E

A H

F

G

E

B

� ∵ BH ⊥ EH and GH ⊥ EH.

∴ ∠BHG is also the angle between

the planes.

P

A

B

Q

R

α

β

T S

R

Q V

W

T S

R

Q V

W

____ is the line of

intersection of

VQST and WRST.

∵ QS ⊥ ____ and

RS ⊥ ____. ∴ ∠______ is the angle

between the planes.

83

11.

The figure shows a cuboid. Name the

angles between the following planes.

(a) Planes ABGF and ADEF

(b) Planes ABHE and CDEH

12.

The figure shows a cuboid. M and N are

the mid-points of FE and GH respectively.

Name the angle between the planes ABNM

and MNCD.

○○○○→→→→ Ex 8D 11, 12

Level Up Question

13. The figure shows a right triangular prism. Name the angle between

(a) the lines PQ and PT,

(b) the line RU and the plane STU,

(c) the planes PQUT and PRST.

B

A

C

D

G

F

H

E

A B

D C

E

F

H

G

M N

P Q

R

T

S

U

84

C

A F

E D

B

H

G

Q

P

R S

C

A D

F

G

B

H

E

3 cm

9 cm

6 cm

A

F

C

B

D

E



Level 1

1. The figure on the right shows a cuboid.

(a) Does each of the following pairs of lines lie on the

same plane?

(i) AB, DC (ii) DE, GF

(iii) GH, CH (iv) FE, AD

(b) For those pairs of lines lying on the same plane in (a),

which of them

(i) are parallel lines? (ii) are perpendicular lines?

2. The figure on the right shows a tetrahedron. Name the line

segment which represents the distance between

(a) point P and line QS,

(b) point Q and line SR.

3. The figure shows a cuboid. Find the distance between

(a) point D and plane EFGH,

(b) point B and plane ADEF.

4. The figure shows a right triangular prism. Name the

angles between the following lines.

(a) BE and ED

(b) AC and BC


8D

85

P Q

R S

T

U V

X P

U

R

Q

S

T

C

A D

F

G

B

H

E

A B

C D

E

F G

H

P Q

A C

E

F

B

D

5. The figure shows a cube. Find the projection of

(a) point S on plane TUVX,

(b) point P on plane QRXV,

(c) point X on plane PQRS.

6. The figure shows a right triangular prism. Find the

projection of

(a) line PQ on plane RSTU,

(b) line RT on plane PUT.

7. The figure shows a cuboid. Name the angle between

(a) line DH and plane EFGH,

(b) line GD and plane CDEH.

8. The figure shows a cube ABCDEFGH. AE and DF intersect at P.

BH and CG intersect at Q. Name the line of intersection of the

planes

(a) ABCD and BCHG,

(b) ADEF and CDEH,

(c) CDFG and EFGH,

(d) ABHE and CDFG.

9. The figure shows a right triangular prism. Name the angle

between the planes

(a) ABFE and ACDE,

(b) DEF and ACDE,

(c) BCDF and ABFE.

86

A

C

E

B

D

G 12 cm

8 cm

16 cm 10 cm

6 cm

F

H

P

Q R

U S

T

X

P

Q

R

U

S

T

F

E

D

A

C

B

Level 2

10. The figure shows a right prism. Find the distance between

(a) point D and line AF,

(b) point G and plane ABCD,

(c) lines EH and FG.

11. The figure shows a cuboid. Find the projection of

(a) line BG on the plane CDEH,

(b) line AH on the plane ADEF,

(c) line EG on the plane ABCD.

12. The figure shows a right triangular prism PQRSTU.

X is a point on QR such that PX ⊥ QR. Name the angle

between line PS and

(a) plane QRSU,

(b) plane PQUT.

13. The figure shows a right hexagonal prism. It is given

that FD ⊥ DC and the plane AFRS is a rectangle.

(a) Name the angle between line FS and plane

(i) ABCDEF,

(ii) CDRS.

(b) Name the angle between planes

(i) FDR and DEQR,

(ii) AFRS and CDRS.

87

14. The figure shows a tetrahedron PQRS, where PQ = PR, QS = RS, PS ⊥ QS and PS ⊥ SR. N is

the mid-point of QR.

Name the angle between planes

(a) PQS and PRS,

(b) PQR and SQR.

15. The figure shows a right prism ABCDEF. Its base is an equilateral triangle. X is the mid-point of

DF.

(a) Amy claims that the angle between line AF and plane CBFD is ∠AFB. Do you agree?


(b) Benson claims that the angle between planes AFD and EFD is ∠AXE. Do you agree?


P

Q

R

S

N

A

B

C

E

F

D

X

88

Answer


1. (a) (i) yes (ii) no

(iii) yes (iv) yes

(b) (i) AB, DC; FE, AD

(ii) GH, CH

2. (a) PS (b) QS

3. (a) 3 cm (b) 9 cm

4. (a) ∠BED (b) ∠ACB

5. (a) T (b) Q (c) R

6. (a) UR (b) UT

7. (a) ∠DHE (b) ∠GDH

8. (a) BC (b) DE

(c) FG (d) PQ

9. (a) ∠BAC or ∠FED

(b) ∠FEA

(c) ∠CBA or ∠DFE

10. (a) 10 cm (b) 16 cm (c) 8 cm

11. (a) CH (b) AE (c) DB

12. (a) ∠PSX (b) ∠SPT

13. (a) (i) ∠SFC (ii) ∠FSD

(b) (i) ∠FDE (ii) ∠FRD or ∠ASC

14. (a) ∠QSR (b) ∠PNS

15. (a) no (b) yes

89

F3B: Chapter 9A

Date Task Progress

Lesson Worksheet


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Book Example 1


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Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)



(Full Solution)


9A Level 1


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90


Self-Test


_________

91


9.1 Pyramids

(a) If a solid has a polygonal base and all its other faces

are triangles with a common vertex, then the solid is

a pyramid.

(b) The figure shows a right pyramid with a rectangular

base. It has the following properties:

(i) NA = NB = NC = ND

(ii) VA = VB = VC = VD

(iii) △VAB, △VBC, △VCD and △VDA are

isosceles triangles.

Note: If the base of a right pyramid is a regular polygon, this right pyramid is called a

regular pyramid.

9.1A Volume of a Pyramid

Volume of a pyramid =

3

1× base area × height


Find the volume of the

triangular pyramid in

the figure.

Find the volume of the

rectangular pyramid in

the figure.

Sol Volume =3

1× 18 × 10 cm3

= 60 cm3

Sol Volume =3

1× ( ) × ( ) cm3

=

○○○○→→→→ Ex 9A 1

height

= 10 cm

base area

= 18 cm2

height

= 6 cm

base area

= 25 cm2

base area

height

92

1. Find the volume of the pyramid with a

square base in the figure.

2. Find the volume of the rectangular

pyramid in the figure.

3. Find the volume of the triangular pyramid

in the figure.

4. The figure shows a regular pyramid with

volume 189 m3. Find the height of the

pyramid.

Let h m be the height of the pyramid.

○○○○→→→→ Ex 9A 2−6 ○○○○→→→→ Ex 9A 9, 11, 12

8 cm

3 cm

7 m

6 m

5 m

13 cm

6 cm 9 cm 9 m

What is the

base

area?

93

9.1B Volume of the Frustum of a Pyramid

(a) When a pyramid is cut by a plane parallel to its base, the removed part at the top is a

small pyramid and the remaining part is called the frustum of the pyramid.

(b) Volume of a frustum of a pyramid

= volume of the larger pyramid − volume of the smaller pyramid


The upper base and the lower base of the

frustum ABCDEFGH in the figure are squares.

Find the volume of the frustum.

The upper base and the lower base of the

frustum ABCDEFGH in the figure are squares.

Find the volume of the frustum.

Sol

Volume of the frustum

= volume of pyramid VABCD −

volume of pyramid VEFGH

=

××−+×× 35

3

1)63(15

3

1 22 cm3

= [675 − 25] cm3

= 650 cm3

Sol Volume of the frustum

= volume of pyramid VABCD −

volume of pyramid __________

=

15 cm A B

C

G

E F

H

D

V

5 cm

3 cm

6 cm 5 m

13.5 m

4 m

6 m

A B

C

G E

F

H

D

V

=

−

94

5. The frustum shown in the figure is formed by cutting

the small right pyramid VEFGH from the large right

pyramid VABCD. The bases of the two right pyramids

are rectangles. Find the volume of the frustum.

○○○○→→→→ Ex 9A 15, 16

10 cm

9 cm A

B

C

G

E

F H

D

V

6 cm

2 cm

3 cm

5 cm

95

9.1C Total Surface Area of a Pyramid

Total surface area of a pyramid

= total area of all lateral faces + area of the base


In the figure, VABCD is a right pyramid with a

square base. Find the total surface area of the

pyramid.

In the figure, VPQRS is a right pyramid with a

square base. Find the total surface area of the

pyramid.

Sol Total surface area of the pyramid

= 4 × area of △VBC +

area of square ABCD

=

+××× 2101210

2

14 cm2

= 340 cm2

Sol

6. The figure shows a regular pyramid

VABCD. Find the total surface area of the

pyramid.

7. In the figure, the base of the right pyramid

VABCD is a rectangle. Find the total

surface area of the pyramid.

○○○○→→→→ Ex 9A 7, 8, 14

Level Up Questions

8. In the figure, VABCD is a right pyramid with a square base.

Its total surface area is 264 cm2. Find the length of a side of

its base.

V

V

V

V

A B

C D lateral

fac

lateral

fac

base

10 cm A

B

C D

V 12 cm

E

15 cm A

B

C D

V 17 cm

E

4 × +

A B

C D

V

20 m

F

13 m

E

32 m

10 m

2 × + 2 × +

Set up an equation

for total surface

area.

15 cm

7 cm

T Q

R P

V S

B

C A

V D

area of the lateral face = 50 cm2

96

9. In the figure, VABCD is a right rectangular pyramid. Find

(a) EC,

(b) VE,

(c) the volume of the pyramid.

(Give the answers correct to 3 significant figures if

necessary.)

(a) In the right-angled triangle ABC,

AC = 22)()( + �Pyth. theorem

=

(b) In the right-angled triangle VEC,

(c)

A

B

C D

V

8 m 15 m

E

13 m

The diagonals of a rectangle

bisect each other. So, EC

=2

1× AC.

97


9 Area and Volume (III)

Level 1

Find the volumes of the following pyramids. [Nos. 1−−−−5]

1. 2. 3.

4. 5.

Find the total surface area of each of the following regular pyramids. [Nos. 6−−−−7]

6. 7.

7 cm

base area = 24 cm2

6 cm

4 cm

5 cm

36 m

The base is a square.

10 m

5 mm

12 mm

9 mm

9 cm

9 cm

area of the

lateral

face

= 60

25 cm

16 cm

16 cm


9A

98

4 cm 4 cm

total surface area = 56 cm2

h cm

Find the unknown in each of the following right pyramids. [Nos. 8−−−−9]

8. 9.

10. The figure shows a pyramid. The length of each side of the square base is 5 cm. If the volume of

the pyramid is 60 cm3, find the height of the pyramid.

11. The figure shows a pyramid PQRS where RS = 6 cm and the height PQ = 7 cm.

If the volume of the pyramid is 28 cm3,

(a) find the area of △QRS,

(b) find the lengths of QR and the slant edge PR.

(Give the answers correct to the nearest 0.1 cm if necessary.)

base area = x m2

20 m

volume = 800 m3

5 P

Q

R

S

7 cm

6 cm

99

12. The figure shows a right pyramid ABCDE, where BC = 30 cm and CD = 14 cm. The heights of

the lateral faces △ABC and △ACD are 20 cm and 24 cm respectively. Find the total surface

area of the pyramid.

13. The frustum shown in the figure is formed by cutting pyramid VPQRS from pyramid VTUXY.

The bases of the two pyramids are rectangles. Find the volume of the frustum.

14. The frustum shown in the figure is formed by cutting pyramid ABCDE from pyramid AFGHI.

The bases of the two pyramids are squares. Find the volume of the frustum.

30 cm

24 cm

20 cm

14 cm

A

B

C

D

E

10 cm

15 cm

6 cm

9 cm

18 cm

V

P

Q R

S

X U

T Y

12 cm

100

12 cm 16 cm

26 cm

A

B

C

D

E

N

Level 2

15. The figure shows a pyramid VPQR whose height is VP. It is given that VQ = 7.5 m, PQ = 6 m

and QR = 10 m. Find the volume of the pyramid.

16. The figure shows a right pyramid ABCDE with a rectangular

base BCDE. The diagonals BD and CE intersect at N. AD = 26 cm,

BC = 12 cm and CD = 16 cm.

(a) Find the lengths of ND and AN.

(b) Find the volume of the pyramid.

17. The cardboard in Fig. A is the net of the right square pyramid VPQRS in Fig. B. It is given that

VN = 30 cm and VQ = 34 cm.

(a) Find the length of PQ.

(b) Find the total surface area of the pyramid.

10 m

V

P

Q

R 7.5 6 m

Fig. A Fig. B

V

R

Q

P

S NV

R Q

P

V

S

N

34 cm

30 cm

V

V

101

A

B

C

D

E

60 cm

50 cm

18. The figure shows a food tent without base. It is

in the shape of a right square pyramid ABCDE,

where BC = 60 cm and AB = 50 cm. Find the

total surface area of the tent.

19. In the figure, VPQRS is a right pyramid whose base PQRS is a rectangle. The height VX of the

lateral face △VPQ is 6.8 cm. PQ = 10.2 cm and QR = 8 cm.

(a) Find the length of VQ.

(b) Find the height of the lateral face △VQR.

(c) Find the total surface area of the pyramid.

20. The figure shows a regular pyramid VABCD. It is given that the area of △VAB is 1 040 cm2 and

the total surface area of the pyramid is 5 760 cm2.

Find

(a) the area of the base ABCD,

(b) the height VQ,

(c) the volume of the pyramid.

10.2 cm

8 cm

6.8 cm

V

Q

R

S

P

X

A

B

C

V

D

Q P

102

21. The figure shows a frustum BCDEFGHI of a right pyramid. Its upper base and lower base are

rectangles with dimensions 10 cm × 7.5 cm and 16 cm × 12 cm respectively. The height of the

frustum is 9 cm. Let h cm be the height of the pyramid AFGHI.

(a) Find the value of h.

(b) Find the volume of the frustum.

22. The figure shows a vessel of capacity 324 cm3. Its shape is an inverted right pyramid whose

base is a horizontal square XYTU of side 9 cm. The vessel contains water to the depth of 8 cm.

(a) Find the height of the vessel.

(b) Find the additional volume of water required to fill up the vessel.

9 cm

16 cm

A

B

C D

E

H I

F G

h cm

12 cm

10 cm

7.5 cm

V

P

Q

R

T

S

U

X

Y

8 cm

9 cm

103

Answer


1. 56 cm3

2. 40 cm3

3. 1 200 m3

4. 90 mm3

5. 800 cm3

6. 321 cm2

7. 1 056 cm2

8. 120

9. 5

10. 7.2 cm

11. (a) 12 cm2

(b) QR = 4 cm, PR = 8.1 cm

12. 1 356 cm2

13. 570 cm3

14. 148 m3

15. 45 m3

16. (a) ND = 10 cm, AN = 24 cm

(b) 1 536 cm3

17. (a) 32 cm (b) 2 944 cm2

18. 4 800 cm2

19. (a) 8.5 cm (b) 7.5 cm

(c) 210.96 cm2

20. (a) 1 600 cm2 (b) 48 cm

(c) 25 600 cm3

21. (a) 24 (b) 1 161 cm3

22. (a) 12 cm (b) 228 cm3

104

F3B: Chapter 9B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)



(Full Solution)


9B Level 1


Teacher’s

Signature

___________

( )


9B Level 2


Teacher’s

Signature

___________

( )


9B Level 3


Teacher’s

Signature

___________

( )


9B Multiple Choice


Signature

___________

( )

105


Self-Test


_________

106


9.2 Circular Cones

(a) A circular cone is a solid enclosed by a circular

base and a curved surface.

(b) (i) If all the slant heights of a circular cone are

equal, the cone is called a right circular

cone.

(ii) By Pythagoras’ theorem,

2ℓ = r2 + h2.

9.2A Volume of a Right Circular Cone

Volume of a circular cone =3

1πr2h

Example 1 Instant Drill 1 Find the volume of the right circular cone in

the figure in terms of π.

Find the volume of the right circular cone in

the figure in terms of π.

Sol Volume =3

1× π × 32 × 6 cm3

= 18π cm3

Sol Volume =3

1× π × ( ) × ( ) cm3

=

3 cm

6 cm

7 cm

9 cm

3

1× base area ×

height

height h slant

height

ℓ

base radius r

height h base radius r

107

1. Find the volume of the right circular cone

in the figure in terms of π.

2. The volume of the right circular cone in

the figure is 940 cm3. Find its height.

( Give the answer correct to 3 significant

figures.)

○○○○→→→→ Ex 9B 1, 2 ○○○○→→→→ Ex 9B 4

Example 2 Instant Drill 2 For the following right circular cone, find

(a) the height,

(b) the volume in terms of π.

For the following right circular cone, find

(a) the height,

(b) the volume in terms of π.

Sol (a) Let h cm be the height

of the cone.

( )2 + ( )2 = ( )2

=

∴ The height of the cone is .

(b) Volume

=3

1× ( ) × ( ) × ( ) cm3

=

Sol (a) Let h cm be the height of the cone.

82 + h2 = 172

h = 22 817 −

= 15

∴ The height of the cone is 15 cm.

(b) Volume =3

1× π × 82 × 15 cm3

= 320π cm3

6 cm

11 cm

10 cm

8 cm

17 cm

20 cm

16 cm

8 cm

17 cm h cm

16 cm

20 cm h cm

108

3. For the following right circular cone, find

(a) the base radius,

(b) the volume, correct to 3 significant

figures.

4. For the following right circular cone, find

(a) the height,

(b) the volume, correct to 3 significant

figures.

○○○○→→→→ Ex 9B 3

9.2B Volume of the Frustum of a Circular Cone

(a) When a right circular cone is cut by a plane parallel to its base, the removed part at

the top is a small right circular cone and the remaining part is called the frustum of

the circular cone.

(b) Volume of a frustum of a circular cone

= volume of larger circular cone − volume of smaller circular cone

12 cm

13 cm 37 cm

24 cm

109


Find the volume of the frustum of the right

circular cone below in terms of π.

Find the volume of the frustum of the right

circular cone below in terms of π.

Sol

Volume of the frustum

= volume of circular cone VAB −

volume of circular cone VCD

=

×××−××× 43π

3

186π

3

1 22 cm3

= (96π − 12π) cm3

= 84π cm3

Sol Volume of the frustum

= volume of circular cone ______ −

volume of circular cone ______

=

5. Find the volume of the frustum of the right circular cone below in terms of π.

○○○○→→→→ Ex 9B 14, 15

6 cm

8 cm 3 cm

V

A B

C D

4 cm

9 cm

5 cm 3 cm

P Q

R S

V

10 cm

= −

20 cm

9 cm 15 cm

A B

C D

V

27 cm

= −

110

9.2C Total Surface Area of a Right Circular Cone

(a) Curved surface area of a right circular cone

= ℓrπ

(b) Total surface area of a right circular cone

= ℓrπ + πr2


Find

(a) the curved surface area, and

(b) the total surface area

of the following right circular cone.

(Give the answers in terms of π.)

Find





Sol (a) Curved surface area

= π × 5 × 13 cm2

= 65π cm2

(b) Total surface area

= (65π + π × 52) cm2

= 90π cm2

Sol (a) Curved surface area

= π × ( ) × ( ) cm2

=


=

13 cm

5 cm

20 cm

12 cm

ℓ ℓ

ℓ

111

6. Find





7. Find





○○○○→→→→ Ex 9B 5, 6

8. In the figure, the curved surface area of a

right circular cone is 24π m2 and the slant

height is 8 m. Find

(a) the base radius, and


of the circular cone.

(Give the answers correct

to 3 significant figures if neccesary.)

9. In the figure, the total

surface area of a right

circular cone is 42π cm2

and the base radius is 3 cm.

Find the slant height of the

circular cone.

Let ℓ cm be the slant height.

(a) Let r m be the base radius.

π × ( ) × ( ) = ( )

=


=

○○○○→→→→ Ex 9B 7, 8, 12

7 cm

16 cm 10 cm

13 cm

3 cm

8 m

112


10. Mary pours 500 cm3 of water into an empty container. The container is in the shape of a right

circular cone of base radius 6 cm and height 13 cm. Will the water overflow? Explain your

answer

Capacity of the container =

∴ The water (will / will not) overflow.

Level Up Questions

11. In the figure, the volume of a right circular cone is 800π cm3 and

its base radius is 10 cm.

(a) Find the height of the circular cone.

(b) Find the total surface area of the circular cone in terms of π.

(a) Let h cm be the height.

(b) Let ℓ cm be the slant height.

12. It is known that the curved surface area of a right circular cone is 369π cm2 and its slant height

is 41 cm.

(a) Find the base radius of the cone.

(b) Find the volume of the cone in terms of π.

Find the slant

height

first.

10 cm

113

h

r



Level 1

Find the volume of each of the following right circular cones in terms of π. [Nos. 1−−−−3]

1. 2. 3.

4. The figure shows a right circular cone of base radius r and height h.

Complete the following table.

r h Volume

(a) 4 m 32π m3

(b) 5 cm 15π cm3

For each right circular cone in Nos. 5–7, find

(a) the curved surface area,

(b) the total surface area.


5. 6. 7.

9 mm

5 mm

45 cm

16 cm

4.5 m 7.5 m

4 cm

8 cm

20

m

33 m

3.6 mm

1.5 mm


9B

114

ℓ

r

8. The figure shows a right circular cone of base radius r and slant height ℓ .

Complete the following tables.

r ℓ Total surface area

(b) (i) 5 mm 80π mm2

(ii) 7 cm 112π cm2

9. The figure shows a tent in the shape of a right circular cone. The curved surface and the base are

made of the same material. The slant height is 2.5 m and the base area is 4π m2.

(a) Find the base radius of the tent.

(b) Find the area of the material used to make the curved surface of the tent in terms of π.

10. The figure shows a right circular cone of height 7.5 cm. If the circumference of its base is

8π cm, find the volume of the cone in terms of π.

11. The base radius and height of a right circular cone are 7 cm and 11 cm respectively. Find the

total surface area of the cone.

(Give the answer correct to the nearest cm2.)

r ℓ Curved surface area

(a) (i) 3 m 12π m2

(ii) 14 cm 70π cm2

2.5 m

7.

115

12. Lucy has eight identical cups in the shape of inverted right circular cones. The base diameter

and the height of each cup are 9 cm and 14 cm respectively. Can the eight cups hold orange

juice of volume 2 250 cm3? Explain your answer.

13. The figure shows a pudding in the shape of the frustum of a right circular cone. The radii of its

upper base and lower base are 6 cm and 7.5 cm respectively. AP = 24 cm and PQ = 6 cm. Find

the volume of the pudding correct to the nearest cm3.

Level 2

14. In the figure, the volume of a right circular cone is 96π cm3 and its height is 8 cm.

(a) Find the base radius and the slant height of the circular cone.

(b) Find the total surface area of the circular cone in terms of π.

15. The figure shows an inverted right circular cone. Its base diameter and curved surface area are

32 cm and 544π cm2 respectively.

(a) Find the height of the circular cone.

(b) Find the volume of the circular cone in terms of π.

7.5 cm

6 cm

24 cm

6 cm

A

P

Q

B C

D E

8 cm

32 cm

116

16. Fig. A shows a glass filled up with apple juice. It is in the shape of an inverted right circular

cone of base radius 9 cm and height 15 cm. All the juice in the glass is poured into an empty

cylindrical vessel of base radius 6 cm (as shown in Fig. B). Find the depth of the juice in the

cylindrical vessel.

17. In the figure, the dimensions of a chocolate cuboid are 12.5 cm × 4 cm × 4 cm. It is melted and

recast into some identical right circular cones of base radius 1.5 cm and height 2.4 cm. What is

the maximum number of cones that can be made?

18. In the figure, a storage tank is composed of a right circular cone and a cylinder with a common

base. The height of the tank is 4 m. The slant height of the cone is 3.25 m and the base diameter

of the cylinder is 6 m.

(a) Find the capacity of the tank.

(b) Find the total curved surface area of the tank.


15 cm

9 cm 6 cm

Fig.

Fig.

4 cm

4 cm 12.5 cm

3.25 m

6 m

4 m

117

19.

The figure shows a bowl. Its shape is a frustum of an inverted right circular cone. The radii of

its upper base and lower base are 10 cm and 7.5 cm respectively. Find the capacity of the bowl

in terms of π.

20.

Fig. A shows a paper cup in the shape of an inverted right circular cone.

(a) Find the capacity of the paper cup.

(b) The paper cup is cut along PQ to form the sector in Fig. B. Find

(i) the area of the sector,

(ii) the angle of the sector.

(Give the answers correct to 3 significant figures.)

48 cm

A

C B

E D

P

Q 10 cm

7.5 cm

20.5 cm 20 cm

Q

P

Q

P

Fig. A Fig. B

O

118

21.

Fig. A shows a sector. It is folded into an inverted right circular cone as shown in Fig. B.

(a) Find the base radius of the cone.

(b) Find the capacity of the cone.

(Give the answer correct to 3 significant figures.)

22.

The figure shows a vessel in the shape of an inverted right circular cone. The vessel contains

some water. The area of the surface of the vessel in contact with water is 252π cm2.

(a) Find the depth of water in the vessel.

(b) Find the additional volume of water required to fill up the vessel.

(Give the answer correct to the nearest cm3.)

12 cm

Fig. B Fig. A

240°

10.5 cm

8.4 cm

O B

A

C

D E P

119

Answer


1. 75π mm3

2. 960π cm3

3. 54π cm3

4. (a) 6 m (b) 3 cm

5. (a) 32π cm2 (b) 48π cm2

6. (a) 330π m2 (b) 430π m2

7. (a) 5.85π mm2 (b) 8.1π mm2

8. (a) (i) 4 m (ii) 5 cm

(b) (i) 11 mm (ii) 9 cm

9. (a) 2 m (b) 5π m2

10. 40π cm3

11. 441 cm2

12. yes

13. 862 cm3

14. (a) base radius = 6 cm, slant height = 10 cm

(b) 96π cm2

15. (a) 30 cm (b) 2 560π cm3

16. 11.25 cm

17. 35

18. (a) 28.5π m3 (b) 26.25π m2

19. 925π cm3

20. (a) 424 cm3

(b) (i) 290 cm2 (ii) 79.0°

21. (a) 8 cm (b) 599 cm3

22. (a) 28.8 cm (b) 2 028 cm3

120

F3B: Chapter 9C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)



(Full Solution)


9C Level 1


Teacher’s

Signature

___________

( )


9C Level 2


Teacher’s

Signature

___________

( )


9C Level 3


Teacher’s

Signature

___________

( )


9C Multiple Choice


Signature

___________

( )


Self-Test


_________

121


9.3 Spheres

(a) The figure shows a sphere.

(i) The point O is called the centre.

(ii) The distance between the centre O and any

point on the surface is the radius (r) of the

sphere.

(b) When a sphere is cut along a plane which passes

through the centre, it is divided into two identical

parts called hemispheres.

9.3A Volume of a Sphere

Volume of a sphere =3

4πr3


Find the volume of the sphere in the figure in

terms of π.

Find the volume of the sphere in the figure in

terms of π.

Sol Volume =3

4× π × 63 cm3

= 288π cm3

Sol Volume =3

4× π × ( )3 m3

=

6 cm 1.5 m

122

1. Find the volume of the sphere in the

figure, correct to the nearest cm3.

2. If the volume of a sphere is 36π cm3, find

its radius.

Let r cm be its radius.

3

4× π × ( )3 = ( )

=

○○○○→→→→ Ex 9C 1(a), 2(a), 3(a) ○○○○→→→→ Ex 9C 4(b), 6

3. The solid in the figure consists of a

cylinder and a sphere. Both of the base

radius of the cylinder and the radius of the

sphere are 6 cm. Find the volume of the

solid in terms of π.

Volume of the solid

= volume of the sphere +

volume of the cylinder

=

4. The solid in the figure consists of a

hemisphere and an inverted right circular

cone with a common base. The radius of

the hemisphere is 12 cm and the height of

the cone is 29 cm. Find the volume of the

solid in terms of π.

Volume of the solid

= volume of the hemisphere +

volume of the cone

=

○○○○→→→→ Ex 9C 11, 12

7 cm

6 cm 10 cm

6 cm

12 cm

29 cm

Volume of a

cylinder =

πr2h

Volume of a hemisphere

=2

1× volume of a sphere

=3

4

2

1× πr3

Volume of a cone

=3

1πr2h

123

9.3B Surface Area of a Sphere

Surface area of a sphere = 4πr2


Find the surface area of the sphere below in

terms of π.

Find the surface area of the sphere below in

terms of π.

Sol Surface area of the sphere

= 4 × π × 52 cm2

= 100π cm2

Sol Surface area of the sphere

= ( ) × ( ) × ( ) m2

=

5. Find the surface area of the sphere below

in terms of π.

6. The surface area of a sphere is 324π cm2.

Find its radius.

○○○○→→→→ Ex 9C 1(b), 2(b), 8

○○○○→→→→ Ex 9C 4(a), 10

5 cm 7 m

16 cm

124

7. Find the total surface area of the

hemisphere below in terms of π.

8. Find the total surface area of the

hemisphere below, correct to the nearest

cm2.

○○○○→→→→ Ex 9C 3(b)

9. The total surface area of a hemisphere is

300π mm2. Find its radius.

10. The curved surface area of a hemisphere is

242π m2.

(a) Find its radius.

(b) Find its volume, correct to

3 significant figures.

○○○○→→→→ Ex 9C 5, 14

6 cm

15 cm

+

curved surface circular

plane

125

Level Up Questions

11. Refer to the figure. A solid metal sphere of radius 6 cm is melted and recast to form a solid

right circular cone of height 13.5 cm. Find the base radius of the circular cone.

Let r cm be the base radius of the circular cone.

Volume of the sphere =

Volume of the circular cone =

∵ Volume of the sphere (= / ≠) volume of the circular cone

∴

12. It is known that the volume of a sphere is 36π cm3. Find the surface area of the sphere, correct

to 3 significant figures.

6 cm 13.5 cm

Suppose the volume of metal

does not change during the

process.

You may keep the

π in the results.

126



Level 1

For each sphere or hemisphere in Nos. 1−−−−3, find the

(a) volume,

(b) total surface area.


1. 2. 3.

4. Find r in each of the following spheres.

(a) (b)

5. Find r in each of the following hemispheres.

(a) (b)

6. The volume of a sphere is 3cm π

3

500. Find its diameter.

7. The figure shows a hemisphere. The area of its base is 900π mm2.

Find its volume in terms of π.

8. The diameter of a spherical Christmas ball is 7 cm. Find the surface area of the ball in

terms of π.

9. The curved surface area of a hemisphere is 200 m2. Find the circumference of its base, correct

to 3 significant figures.

6 cm

18 mm

3 m

r cm

surface area = 100π cm2

r m

volume =3m π

3

4

r mm

curved surface area = 98π mm2

r cm

total surface area = 12π cm2


9C

127

10. The figure shows a toy. It consists of a right circular cone and a hemisphere with a common

base. Find the total surface area of the toy.

(Give the answer in terms of π.)

11. The surface area of a spherical basketball is 1 850 cm2. Find

(a) its radius,

(b) its volume.

(Give the answers correct to 3 significant figures.)

Level 2

12. The volumes of the sphere and the cylinder in the figure are the same. Find the base radius of

the cylinder.

13. In the figure, there is some water in a cylindrical container of base radius 4.8 cm. An iron

sphere of diameter 7.2 cm is immersed in water. Then, the depth of water rises to 13 cm. Find

the original depth of water.

14. A large solid metal sphere of radius 15 cm is melted and recast into 27 identical small solid

metal spheres. Find the surface area of each small sphere in terms of π.

6 cm 14 cm

9 cm

6.75 cm

13 cm

4.8 cm

128

15. The figure shows a steel bowl in the shape of a hemisphere. The bowl is 0.2 cm thick with an

internal diameter of 15.6 cm. If there is 600 cm3 of steel, at most how many bowls can be made?

16. The solid in the figure is formed by two hemispheres. The radius of the smaller hemisphere is

4 cm. The curved surface area of the larger hemisphere is 40.5π cm2. Find the total surface area

of the solid, correct to the nearest cm2.

17. The figure shows a right circular cone of base radius 14.4 cm and a hemisphere of radius

12 cm. Their total surface areas are the same.

(a) Find the slant height of the cone. (b) Is the volume of the hemisphere greater than that of the cone? Explain your answer.

18. In the figure, a vessel consists of a hemisphere and a cylinder with a common base. The base

diameter of the hemisphere is 30 cm. The capacity of the vessel is 9 000π cm3. Is the height of

the vessel less than 40 cm? Explain your answer.

0.2 cm

15.6 cm

4 cm

12 cm 14.4 cm

129

19. The solid in the figure consists of a right circular cone and a hemisphere with a common base.

The height of the cone is 24 cm. The ratio of the volume of the cone to the volume of the

hemisphere is 2 : 3.

(a) Find the radius of the hemisphere.

(b) Find the volume of the solid in terms of π.

(c) Find the total surface area of the solid in terms of π.

24 cm

130

Answer

Consolidation Exercise 9C

1. (a) 288π cm3 (b) 144π cm2

2. (a) 972π mm3 (b) 324π mm2

3. (a) 18π m3 (b) 27π m2

4. (a) 5 (b) 1

5. (a) 7 (b) 2

6. 10 cm

7. 18 000π mm3

8. 49π cm2

9. 35.4 m

10. 132π cm2

11. (a) 12.1 cm (b) 7 480 cm3

12. 12 cm

13. 10.3 cm

14. 100π cm2

15. 7

16. 241 cm2

17. (a) 15.6 cm (b) yes

18. no

19. (a) 18 cm (b) 6 480π cm3

(c) 1 188π cm2

131

F3B: Chapter 9D

Date Task Progress

Lesson Worksheet


(Full Solution)



(Full Solution)


9D Level 1


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9D Multiple Choice


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Self-Test


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132


9.4 Dimensions of Length, Area and Volume

Measurement Dimension

Example:

Cube

Linear (involving the lengths of

line segments or curves, the sum

or multiple of lengths)

1 Perimeter of a face = 4x

Quadratic (involving the product

of two linear measurements) 2 Total surface area = 6x2

Cubic (involving the product of

three linear measurements) 3 Volume = x3

Consider the rectangular pyramid in the figure. For each expression in the table, indicate its kind of

measurement and write down its dimension. [Nos. 1– 4]

Expression Linear Quadratic Cubic Dimension

1. h

2. ab

3. abh

4. 2(a + b) ○○○○→→→→ Ex 9D 1−3

The formulae of two quantities in the following solids are given. Determine whether the quantity

represented by each formula can be the total surface area or the volume of the solid. [Nos. 5–6]

5. Regular pyramid

Formula (a): X =3

1a2h

Formula (b): Y = a2 + 2ab

x x

x

a

b

h

Degree of a1b1

= 1 + 1 =

( )

� Degree of

expression = 1

� Degree of

expression = 2

� Degree of

expression = 3

Degree of

a1b1h1

=

Find the degree of

the expression

first.

h

a

b

133

6. Cylinder

Formula (a): X = 2πr(r + h)

Formula (b): Y = πr2h

○○○○→→→→ Ex 9D 4, 5

Level Up Questions

Refer to the solids as shown. Identify which measurement (linear, quadratic or cubic) is represented

by each of the expressions below. [Nos. 7–8]

7. (a) h + r

(b) 3πr2 + 2πrh

(c) πr2h +3

2πr3

8. (a) Ah

(b) 2(x + h)

(c) 2A + 4xh

r h

h

x

area = A

r h

Note that A represents

a quadratic

measurement.

134

a

b c

x

x y

d

z

r

ℓ

a

b x

y h



Level 1

Refer to the solids as shown. Write down the dimensions of the quantities represented by the

expressions below. [Nos. 1–3]

1. (a) a + b + c

(b) bc

(c) abc

2. (a) yx2

2

1

(b) 2(x + y)

(c) yxx )22( +

3. (a) zdπ

(b) 2

d

(c) zdd23

π4

1π

12

1+

The formulae of two quantities in the following solids are given. Determine whether the quantity

represented by each formula can be the total surface area or the volume of the solid. [Nos. 4–5]

4. Right rectangular pyramid

Formula (a): abhP3

1=

Formula (b): Q = ax + ab + by

5. Half-cylinder

Formula (a): ℓrrX )2π(π2 ++=

Formula (b): ℓ2

π2

1rY =


9D

135

h

a

a

b

c

A

Level 2 Refer to the solids as shown. Identify which measurement (linear, quadratic or cubic) is represented

by each of the expressions below. [Nos. 6–7]

6. (a) 4a

(b) ha2

3

1

(c) 224 aha +

7. (a) Ab

(b) 22 bc −

(c) 2262 bcbA −+

8. Let a, b and c be the linear measurements of a solid. Carman and Dickson express the volume of

the solid in terms of a, b and c as a2b + bc2 – c3 and 4ab + 3bc + 2ac respectively. It is known

that one of the expressions must be wrong. Whose expression is it? Explain your answer.

(A refers to the base area of the prism.)

136

Answer


1. (a) 1 (b) 2 (c) 3

2. (a) 3 (b) 1 (c) 2

3. (a) 2 (b) 1 (c) 3

4. (a) volume

(b) total surface area

5. (a) total surface area

(b) volume

6. (a) linear measurement

(b) cubic measurement

(c) quadratic measurement

7. (a) cubic measurement

(b) linear measurement

(c) quadratic measurement

8. Dickson

137

F3B: Chapter 9E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)

Book Example 20


(Video Teaching)



(Full Solution)


9E Level 1


Teacher’s

Signature

___________

( )


9E Level 2


Teacher’s

Signature

___________

( )


9E Level 3


Teacher’s

Signature

___________

( )

138


9E Multiple Choice


Signature

___________

( )


Self-Test


_________

139

Book 3B Lesson Worksheet 9E (Refer to §9.5)

9.5A Similar Plane Figures

For two similar plane figures,

if A1 and A2 denote their areas,1ℓ and

2ℓ denote any two corresponding lengths,

then

2

2

1

2

1

=ℓ

ℓ

A

A.


The figure shows two similar triangles A and

B. Find the ratio of the area of A to that of B.

The figure shows two similar rectangles P and

Q. Find the ratio of the area of P to that of Q.

Sol B

A

of Area

of Area =

2

cm6

cm10

=

2

3

5

=9

25

Sol Q

P

of Area

of Area =

)(

cm)(

cm)(

=

1. The figure shows two similar hexagons R

and S. Find the ratio of the area of R to that

of S.

2. Figures M and N as shown below are

similar. Find the ratio of the area of M to

that of N.

○○○○→→→→ Ex 9E 1, 2

6 cm 10 cm

A B 6 cm 3 cm P Q

9.6 cm 8 cm

R S

7 cm 14 cm

M N

140

3. The ratio of a pair of corresponding sides

of the two similar triangles as shown

below is 2 : 3. If the area of the large

triangle is 162 cm2, find the area of the

small triangle.

Let x cm2 be the area of the small triangle.

)(

2

2

)(

2

cm )(

cm

=

x

=

4. Figures C and D as shown below are

similar. If the area of figure C is 176 cm2,

find the area of figure D.

○○○○→→→→ Ex 9E 3−6


The actual area of a playground is 1 000 m2,

and its area on a plan is 2.5 cm2. If the scale of

the plan is 1 : k, find the value of k.

The actual area of a park is 15 km2, and its area

on a plan is 0.6 m2. If the scale of the plan is

1 : h, find the value of h.

Sol 5.2

1001000001 ×× =

2

1

k

4 000 000 = k2

k = 2 000

Sol )(

)()(15 ×× =

)(

)(

)(

=

5. On a floor plan of scale 1 : 120, the area of

a shop is 6 cm2. Find the actual area of the

shop in m2.

Let x m2 be the actual area of the shop.

)(

)()( ××x=

)(

)(

)(

=

6. On a map of scale 1 : 10 000, the area of

Victoria Park is 19 cm2. Find the actual

area of Victoria Park in m2.

○○○○→→→→ Ex 9E 13

area

= 162 cm2 area = 176 cm2 4 cm 5 cm

C D

2

2

1

2

1

=ℓ

ℓ

A

A

1 m = 100 cm

1 m2

= 100 × 100 cm2

1 km = 1 000 m

1 km2

= ( ) × ( ) m2

141

9.5B Similar Solids

For two similar solids,

if V1 and V2 denote their volumes, A1 and A2 denote the areas of any two corresponding

surfaces, 1ℓ and

2ℓ denote any two corresponding lengths, then: 2

2

1

2

1

=ℓ

ℓ

A

A and

3

2

1

2

1

=ℓ

ℓ

V

V


The figure shows two similar solids A and B.

Find the ratio of

(a) the total surface area of A to that of B,

(b) the volume of A to that of B.

The figure shows two similar solids C and D.

Find the ratio of

(a) the total surface area of C to that of D,

(b) the volume of C to that of D.

Sol (a) B

A

of area surface Total

of area surface Total =

2

3

5

=9

25

Sol (a) D

C

of area surface Total

of area surface Total =

)(

)(

)(

=

(b) B

A

of Volume

of Volume =

3

3

5

=27

125

(b) D

C

of Volume

of Volume =

)(

)(

)(

=

7. The figure shows two similar solids P and

Q.

Find the ratio of

(a) the total surface area of P to that of Q,

(b) the volume of P to that of Q.

8. The figure shows two similar solids R and

S.

Find the ratio of

(a) the curved surface area of R to that of S,

(b) the volume of R to that of S.

○○○○→→→→ Ex 9E 7, 8

9 cm 6 cm

P Q

12 cm R S

15 cm

3 cm 5 cm A B

2 m 1 m C D

142

9. The heights of the two similar cylinders as

shown below are 8 cm and 12 cm. If the

total surface area of the small cylinder is

28 cm2, find the total surface area of the

large cylinder.

10. The heights of the two similar prisms as

shown below are 21 cm and 15 cm. If the

volume of the large prism is 171.5 cm3,

find the volume of the small prism.

Let x cm2 be the total surface area of the

large cylinder.

2

2

cm )(

cm )(=

)(

)(

)(

=

11. The ratio of the heights of two similar

circular cones is 1 : 5. If the curved surface

area of the large cone is 100 cm2, find the

curved surface area of the small cone.

12. The ratio of the lengths of a pair of

corresponding sides of two similar solids

is 4 : 3. If the volume of the small solid is

189 cm3, find the volume of the large

solid.

○○○○→→→→ Ex 9E 9−12

8 cm 12 cm 15 cm 21 cm

2

2

1

2

1

=ℓ

ℓ

A

A

3

2

1

2

1

=ℓ

ℓ

V

V

143

13. In the figure, the pyramid VABCD is divided by a plane parallel to its

base into two parts VEFGH and ABCDEFGH. It is given that

BC : GH = 3 : 1.

(a) If the area of quadrilateral ABCD is 63 cm2, find the area of

quadrilateral EFGH.

(b) If the volume of VEFGH is 21 cm3, find the volume of VABCD.

○○○○→→→→ Ex 9E 14

Level Up Question

14. In the figure, there is some water in a vertical right conical vessel.

The area of the surface in contact with the water is

4

1 of that of the

curved surface inside the vessel.

(a) Find

2

1

r

r.

(b) If the volume of water in the vessel is 17 cm3, find the capacity

of the vessel.

A B

C

G

E

F

H

D

V

r2 r1

If a small pyramid is removed

by cutting a large pyramid

along a plane parallel to

its base, the part removed

is similar to the original

solid.

144



Level 1

In each of the following, figure A and figure B are similar. Find the ratio of the area of A to that of B.

[Nos. 1–2]

1. 2.

Find the unknown in each of the following pairs of similar plane figures. [Nos. 3–5]

3. 4.

5.

In Nos. 6–7, solid A and solid B are similar. Find the ratio of

(a) the curved surface area of A to the curved surface area of B,

(b) the volume of A to the volume of B.

6. 7.

A

B

3 cm

2 cm A

B

8 cm

10 cm

area = 2.7 cm2

5 cm 3 cm

area = T cm2

area = P m2

area = 324 m2

28 m

36 m

32 cm

area = 16 cm2 area = 9 cm2

x cm

7 m

3.5 m A B

15 cm

18 cm

B A


9E

145

scale 1 : k

Find the unknown in each of the following pairs of similar solids. [Nos. 8–11]

8. 9.

10. 11.

12. The figure shows a map of an island of scale 1 : k. The area of the

island on the map is 9 000 cm2 and the actual area of the island is

3 600 000 m2. Find the value of k.

13. In the figure, the pyramid VXYZ is divided by a plane parallel to its base into two parts VPQR

and PQRZXY. It is given that VX = 2VP.

(a) If the area of △PRQ is 85 cm2, find the area of △XZY.

(b) If the volume of pyramid VXYZ is 3 200 cm3, find the volume of pyramid VPQR.

4 cm

total surface area

= 200 cm2

total surface area

= x cm2

3 cm 15 m 9 m

volume = 24 000 m3 volume = v m3

h cm

3 cm

base area

= 36 cm2

base area

= 81 cm2

x mm

1 mm

volume = 125 mm3 volume = 64 mm3

V

P Q

R

X Y

Z

146

A

B

C

E

D

R Q

P S

Level 2

14. In the figure, ABCDE and ASRQP are two similar pentagons. If

AB : AS = 2 : 3, find the ratio of the area of pentagon ABCDE to

that of polygon BCDEPQRS.

15. In the figure, ABE and ACD are straight lines. BC // ED and AC : CD = 3 : 1. It is given that the

area of △ADE is 96 cm2. Find the area of the quadrilateral BCDE.

16. The curved surface areas of two similar cylinders P and Q are 810 cm2 and 250 cm2

respectively.

(a) Find the ratio of the base diameter of P to that of Q.

(b) A student claims that the volume of P is 6 times that of Q. Do you agree? Explain your

answer.

17. The figure shows two similar rectangular bars. The total surface area of the shorter bar is 64%

of that of the longer bar. If the volume of the shorter bar is 320 cm3, find the volume of the

longer bar.

A

B C

D E

147

18. The figure shows two dolls which are similar solids. The ratio of the volumes of the two dolls is

27 : 125.

(a) Find the height of the smaller doll : the height of the larger doll.

(b) If the area of the face on the larger doll is 150 cm2, find the area of the face on the smaller

doll.

19. In the figure, a right circular cone P is divided into a frustum X and a right circular cone Y. The

cones P and Y are similar. The heights of X and Y are 6 cm and 24 cm respectively.

(a) Find the volume of the frustum X : the volume of the cone Y.

(b) If the curved surface area of Y is 400 cm2, find the curved surface area of X.

20. The figure shows a right rectangular pyramid VTUXY. It is cut by a plane parallel to its base into

pyramid VPQRS and frustum PQRSTUXY. It is given that

area of △VPS : area of quadrilateral PSTU = 4 : 5. If the volume of pyramid VTUXY

is 1 080 cm3, find the volume of frustum PQRSTUXY.

6 cm

P

24 cm

X

Y

U

V

T

X Y

P S

Q R

148

Answer

Consolidation Exercise 9E

1. 4 : 9

2. 16 : 25

3. 7.5

4. 196

5. 24

6. (a) 4 : 1 (b) 8 : 1

7. (a) 25 : 36 (b) 125 : 216

8. 112.5

9. 5 184

10. 4.5

11. 1.25

12. 2 000

13. (a) 340 cm2 (b) 400 cm3

14. 4 : 5

15. 42 cm2

16. (a) 9 : 5 (b) no

17. 625 cm3

18. (a) 3 : 5 (b) 54 cm2

19. (a) 61 : 64 (b) 225 cm2

20. 760 cm3

Chapter 7 Quadrilaterals

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