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(Unless 0therwise specified, numericaJ answers should be either exact or correct to 3 signific·ant figures.)

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1·The figure shows a cube of side 1 cm. Find the angles between
(a) the lines AF and FG,
E
(b) the lines AH and H C,
(c) the line AF and the plane ABCD,
(d) the planes AHD and ABCD.
A 1 cm

2·The figure shows a rectangular block with dimensions 6 cm x 3 cm x 4 cm. B A

~
II A.'!
Find
(a) the lengths of AG and EG, / / / l'+Cm
c/
丨
(b) the shortest distance between the point A and the plane EFGH, G
'
'' '
(c) the shortest distance between the point F and the line EG.

H
I,
' 6cm
、V
E

3. The figure shows a wooden rectangular board ABCD which is 50 cm long ^

,o
and 40 cm wide. It is inclined to the horizontal plane BCEF at 10°. Find I
I
the angles between H I
jE
(a) the lines BD and BA,
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{b) the line BD and the horizontal plane BCEF.
B 50cm

4. In the figure, ABCDHGFE is a rectangular block with dimensions H E

5 cm x 12 cm x I6 cm. The diagonals of rectangle GFEH intersect at P.

'
Find C
(a) the shortest distance between the point P and the line AB, G
u-: \~
(b) the angle between the line PA and the base ABCD. 5cm
匕之．
A 12 cm B

5. In the figure, ABCFDE is a right triangular prism of height 8 cm and A

its base is an equilateral triangle of side 8 cm. Xis a point on AD \
C
such that the angle between the planes XEFand DEF is 36°. Find the
length of AX.
8cm

E F
8cm

'
,
tJJe figure, PA and ABC are the h·
- 醇llcations ofTrlyonometry in 3-dimensional Prob/ems

E,̀ JJl 懿edron PABC respective! eight and the b

P r
U
碰 Y. Ir 乙p BA == 3Qo ase of the
72°and BC== 14 cm, find' 乙ABC :: 600, -
瓦＇：：：：：： c
-C
the lengths of AB and PA, .,
(ll) c
the area of D.ABC, -,
(b)
the volume of the tetrahed
(c) ron PABc. B

c
Jo the figure, a thin metallic sheet PQR .
1. vertical stick PD so that D, Q and R / supported by
a le on the same
hO「izontal ground. 6.PQR is an isosceles t
nangle with
,。：：：： PR, QR= 120 cm and 乙 QPR:::: 30°. I
that 即== 210 cm. Find tis also given
闐創団

the length of PR,

the length of the stick PD,
the angle between the th·lil metallic sheet p
尸

QR and the horizontal ground.

for each of the following figures,
8,
(i) find the length of the longest line se
gment formed by joining any two vertices,
(ii) fmd the angle between th·
e lme segment in (i) and the plane ABCD.
D
(a) 3m
(b) 6cm D

'
I
I 5m
B

I
I
E)- - _, 一一 H
F ,,
G V
a cuboid a right pyramid
二

T
9, A building TP of height 45 m stands vertically on a level ground. The
bearings of two points, A and B, on the ground from the building are N
150°and 200°respectively as shown in the figure. The angles of elevation
of T from A and B are 30°and 40°respectively. Find 5
°3
o

(a) the distance between A and B, ,`: ,4

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/V -
丶

B
Ifg

(b) the true bearing of A from B.

丶

0
。
、、
、

、3

`A
丶
、

10. In the figure, A, B, C and D are points on the horizontal ground

D
for面ng a rectangle with dimensions 3 m x 2 m. PB is a vertical flag
at B. The compass bearing of A from B is NI 5°W and the angle of
elevation of P from A is 50°. Find
因徇的

the height of the flag PB,

the compass bearing of P from D,
門

the angles of elevation of P from

m C, (ii)
D
 Measures, Shape and Space

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;~80『
a constant height 800 m
11. An aeroplane is flying due east at
the horizontal ground.
above the ground. C is a point on
the angle of elevation of the
Initially, a man at C observes that
onds, the man finds that the E
aeroplane at A is 50°. After 10 sec
°. If A is due north of C
bearing of the aeroplane at Bis 070
stant speed, find
and the aeroplane is flying at a con
(a) the speed of the aeroplane in mis,
(b) the angle of elevation of B from C.
on the same
a vert ·1 stick PT so that Q, R and T lie
1ca
R is supported by
12. In the fi郡n; a triangular board PQ
= 21 cm, PT = 10 cm and 乙PQR=
36°.
horizontal table. PQ = 20 cm, QR
P

.
(a)Find the area of L:,.PQR.
.
(b) Hence, find the shortest distance bet
ween the point T and the line QR
v
VA BCD with a square base of side
13. The figure shows a right pyramid 0.
the line VA and the base ABCD is
8 cm. Suppose the angle between
(a)
m
Find the length of AC and leave
your answer in surd
c
form.
pyram id in terms of 0.
(ii) Hence, express the height of the

(b) If the volume of the pyramid

is 144 cm3, find the value of 0.

•
A B
8cm

a
It is given tha t a reg ula r pen tagon ABCDE of side 5 cm. 0 is
14. (a)
t OA =O B= OC =O D= OE.
poi nt inside the pentagon such tha
Find the length of OA.
sists of
(b) The figure on the right
shows a net for a pyramid. It con
five identical isosceles triangles
the pentagon mentioned in (a) and
. Fin d the volume of the right
of sides 12 cm, 12 cm and 5 cm
pyramid formed. D
A
l
ABCD is inclined to the horizonta
15. In the figure, the rectangular plane vely.
II'

und BC EF at 20°. F and E are verticaily below A and D respecti E

gro with F' ---
the plane AB CD . The path CP makes an angle 50° ,
CP is a pat h on '
the line of greatest slope CD of the
inclined plane.
,' '
(a) Express CD in terms of DE. B

plane BCEF.
(b) Hence, find the angle between CP and the

5
 e shows a tetrahed ronDABc
囯gur
60 crn, AC= 65 cm, DA::::: 叭th AB ::::

'
Jb, pC:;:; 16 crn , DB ::::: l 20 叩， 65 cm
, C:;:; 63 cm. 2 cm and
c
Is LBDC the angle betWeen th
(9) p;xplain your answer. e planes ADB and ADC?
e 12cm
Is LADB the angle between th
(b) e planes CDB and CDA?
p;xplain your answer.

the figure, ABCD is a rectangular·

Jo inclined c
,1. d AD == 4 m slopmg at an angle of 300 plane with AB= 5 m
。叩d. His a point on CD. E, Kand F 叭th the horizontal
aO
gr are the projections E
f C, Hand D on the horizontal gro皿d
。given that 乙HAK= 26°and 乙HBK == 20~~spectively. It is
A
(a) (i) Find the lengths of AH and BH. 26° 20°
(ii) Hence, find 乙AHB.

•: A small blue toy car goes straight from A

toy car goes straight from B to H to H at an average speed of 0.3 mis, while a small green
at an average speed of 0.4 mis. The two toy cars start moving at
the same tlllle. Will theY reach H at th
e same time? Explain you answer.

~
VABCD is a right pyramid with v
]8. In the figure, a square base of side 1Ocm.
The diagonals AC and BD intersect at o and VO= 8cm.
(a) Find
(i) the length of VB,
(ii) 乙 VBA.

(b) If N is a point on VB such that AN .l VB, find the length of

A
AN. 10 cm
B

(c) Hence, find the angle between the planes VAB and VBC.
A
19. In the figure, ABCD is a tetrahedron. 6-BCD is the base where
BC= BD = 8 cm and CD = 6 cm. The height AB of the tetrahedron is
10 cm.Mand N are the mid-points of AC and AD respectively. Find
10cm
the angles between
(a) the planes ACD and BCD,
(b) the planes AMN and BMN.

20. In the figure, rectangle ABCD is a part of an inclined plane. D

ABCD inclines at 25°to the horizontal ground BCFE. A and D

are vertically above E and F respectively. A man at C walks 10 m
along CD to X, straight across the road to Yand walks 10 m
30/m/, F

along BA to A. If AB= 30 m and AD= 15 m, fio<l

'
(a) the total distance travelled by the man,
horizontal ground. B c
(b) the angle between the path XY and the 6.S:
 uo l 21. ln the fi严 AB is a vertical pole of height 4 m. C is a point IO m
一
lepuno,

due east of Band AC is a tightened rope. The sun shines from

,UON N30°W at an angle of elevation 60°and the shadow of AB and AC
on the level ground are FD and FC respectively.
(a) Find E

(i) the distance between Band F

(ii) tbe distance between C and F,
(iii) 乙FAC.
｀蕊
(b) William claims tbat tbe length of tbe shadow of AC will decrease if the sun shines at a greater angle
of elevation (which is smaller than 90°). Do you agree? Explain your answer.
A
22. In tbe figure, AH and BK are two towers 600 m apart. K is due east
一－
of H. H, Kand P are pomts on tbe horizontal ground. Tbe compass - B
bearings of Hand K from Pare N65°W and N40°E respectively. ``

The angles of elevation of A and B from Pare 44°and 33°

respectively. E
H 、,
(a) Find the distances between
(i) Pand H,
..'-'!, j守,'
丶 ,65° / 今、
40°
(ii) P and K.

(b) Find the heights of the towers AH and BK.

(c) Find the angle of depression of B from A.

23. A wall of height 4 m stands on the horizontal ground and lies in the
east-west direction. A triangular plastic sheet ABC is mounted on
the wall such that BC lies along the top of the wall and A is
supported by a vertical post of height 5 m. The three sides of the
sheet ABC are 7 m, 7 m and 10 m with BC being the longest side. At E
noon, the sun shines from the top vertically and casts a shadow
!::,PQR on the ground.
(a) Find the angle that the sheet makes with the vertical wall. P

(b) Find the area of the shadow of the sheet at noon.

｀桑
(c) In the evening, the sun shines from the west at a certain angle of elevation. Gloria claims that the
area of the shadow is less than the area of 11PQR. Do you agree? Explain your answer.
De`,
24. In the figure, 0, A and Bare three points on a straight line such that ,m
'
丶

``
丶

OA = 60 m and OB= 90 m. 0, A, Band C lie on the same level. D

丶

/h
'
丶

`
丶

is vertically above C such that DC= h m. The angles of elevation of '

丶

'
丶

D from 0, A and Bare 30°, 60°and 45°respectiv ely. ' I

' B
户 1350
(a) By considering !::.COA, show that cos 乙 COA= .
45./3h
h2+4050
(b) By considering !::.COB, show that cos 乙 COB= . 0~90m
90 出
(c) Hence, find the value of h.
(d) Given that C is due north of O, find the compass beanng of B
from 0.
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Appl/cations~伊l9onometry In
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(a) shows a triangular cardb oard Non

figure
p· pf),:::; 53 cm, CD= 100 cm and AD ::::: 40 ::_c. D is a point on BC such that AD is perpendicular to BC -
foun

A A

Figure (a)
c
` Figure (b)
find the lengths of AB and Ac.
(11)
The cardboard is then folded al ongAD and
(b) put on a horizontal table with BD and CD lying on it as
shown in Figure (b). It is given that
the distance between Band Con the horizontal table is 14 l cm.
(i) Find the area of MBC.

(ii) Hence, or otherwise, fmd the shortest d'

1stance between D and the plane ABC.
f igure (a) shows a nght-angled triangular cardboard AEC, where AB= BC= 5 cm, CE= 6 cm and
26, LA DE= 乙AEC= 90°.
C

6cm ..
A

Figure (a) Figure (b)

(a) Find the length of AD.
(b) The cardboard is then folded along BD such that 乙ADE= 40°. It is then put on a horizontal table
with AD and DE lying on it as shown in Figure (b). Find
(i) the angle between the line AC and the horizontal table,

(ii) the area of~ABC.

8 cm.
27. Figure (a) shows a rectangular block ABCDEFGH with height 6 cm and a square base of side
Atetrahedron ABDFis cut off from the block and the remaining solid is shown in Figure (b).
D

A
6cm
.. 6cm

F
8cm 8cm
F
Figure (a) Figure (b)

If Mis the mid-point of DB, find the angles between

(a) the lines DF and BF,
(b) the line FM and the plane EFGH,
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6

(c) the planes BDF and BCD.

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6cm- C
B,,- ---
B 6cm c
Figure (b)
Figure (a)
(a) Find the length of CQ.
OWs

(b) Let a be the angle between

the planes ACQP and ACD.

(i) Find a.
you
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t a is greater than the angle between the line QC and the plane ACD. Do
(u) Tom claims tha
agree? Explain your answer.
F
ABCDEFGH with a square base
30. The figure shows a crystal souvenir
EFGH is an inclined plane in the
ABCD lying on a horizontal table.
and Ha re vertically above D, A, B
shape of a rhombus, where E, F, G
AB = 8 cm, AF = 24 cm,
and C respectively. It is given that 24c m
H
.
BG = DE = 18 cm and CH = 12 cm
18
(a) Find the length of GH. 12cm
A
(b) Find the area of the rhomb
us EFGH.
｀哀 Can a piece of circular sticker of radius 5.5 cm be stuck onto
the
(c) B
circular sticker completely lies
inclined surface EFGH so that the
answer.
in the region EFGff? Explain your
 re (a) shows a tnangular Paper card ABC Non

囹 oiJit of BC. The paper card . h AB=

faidedWit
,found
the
iswn n AC= 25 cm and BC= 14 cmnta .D is the
al
訌;4B and AC lying on it as sho .inp ·
JI' 謔 ong the line AD and put on a horizo l table
A ) . Let
唧 re (b
乙BDC=0.
D

B D c
.
I<

Refer to Figure (b).

14 cm
Figure (a)
- Figure (b)

河pose 0 = 30°, fmd

(a)
(i) the area of 6.BDC,
(ii) the volume of the tetrahed ronDABC.
｀亮
(b) Describe how the volume of th e tetrahed ron DABC varies when 0 increases from 30 °to 150°.
Explain your answer.

= 60 °an d the
a rhombus, where AB = 12 cm, 乙ABC
In Figure (a), ABCD is a cardboard
in the shape of
32,
diagonals AC and BD intersect at M. The cardboard is folded along AC such that 乙BMD =
t:,ABC lies on the horizontal PIane as shown in Figure (b).
A
口 30 °an d

.. \ :
y
B D

c
C
Figure (a) Figure (b) Figure (c)
ne ABC.
(a) Find the height of D above the pla
the plane ABC.
(b) Find the angle between AD and
严
c) If the cardboard is folded along
BD such that 乙 CMA = 30° as shown in Figure (c), is the
angle
(b)? Explain yo ur
t in
al to, less than or greater than tha
between BC and the plane ABD equ
answer.
 110可per
•-Measures;--:,napec1 ~"'~

CD = 13 cm an d
AD = 24 cm, BC =
per card ABCD with AB =
33. Figure (a) shows a piece of pa
乙BAD=60°.
D c

..
B
Figure (b)
Figure (a)

(a) Fin d the length of AC

. as shown in
d AD lie on a horizontal table
t AB an
(b) The pa pe r card is the
n folded along AC such tha
AD = 30°. Fin d
Figure (b). It is given that 乙B
table,
line AC and the horizontal
(i) the angle between the
edron CARD.
(ii) the volume of the tetrah
is du e east of A
s on the ho riz on tal gro un d where AB = 30 m an d B
po int tal ground. When
In the figure, A an d B are AD an d BC vertical to the ho riz on
pe of a tra pez ium wi th
ABCD is a bo ard in the sha on the horizontal
an an gle of ele vat ion 40 °, the sha do w of the wall
with
the su n shines from N5 0°W
m.
tha t AD = 12 m an d BC = 5
gro un d is AB GF It is given

1 sun ray

` N

BG.
(a) (i) Fin d the lengths of AF an d
the sha do w AB GF
(ii) He nc e, find the are a of
the area of
琴
wi th an an gle of ele vat ion 60°. Ph ilip claims tha t
(b) Suppose the sun 洳 nes
from N5 0°W you
gro un d is gre ate r tha n the are a ob tai ne d in (a). Do
the ho riz on tal
the shadow of the wall on
agree? Explain yo ur answer.
claims
｀桑 an an gle of ele vat ion 40 °, where 0° < 0 < 50°. Sa m
from N0 W with
(c) Suppose the sun shines ho riz on tal gro un d 1s gre ate r tha n the are
a obtained in
do w of the wa ll on the
tha t the area of the sha
ur answer.
(a). Do you agree? Explain yo

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