Exercise 1 : Based on the diagram, calculate the angle between the line and the plane given
Lines and Planes in 3-Dimensions
2
U
a) The diagram shows a cuboid. Calculate the angle between line NE and the plane of GFKN K
b) The diagram shows a cuboid with with a horizontal base JKLM .Calculate the angle between line KS and the plane of SRLM. R S
L
12 cm N
P
M F
E
Q L
M
8 cm
5 cm G
16 cm
5 cm
H
J
c) The diagram shows a prism. Calculate the angle between line line RY and the plane of STY. X
K
6 cm
d) The diagram diagram shows shows a prism. Calculate Calculate the angle between line line QE and the plane of of DCE.
E
Y
5 cm
6 cm
F
T
U 8 cm R
14 cm
P
C
12 cm
S
A
e) The diagram shows a pyramid . Given that HP = 13 cm. Calculate the angle between line
Lines and Planes in 3-Dimensions
D
Q
6 cm
B
f) The diagram shows shows a prism. Calculate the angle between line UV and the plane of PSWV.
3
PG and the plane of EHP.
W
P V
7 cm H
G
X
3 cm
U S
R
4 cm 9 cm E
P
F
7 cm
g) The diagram shows shows a pyramid with a horizontal base DEFG. Given that VO = 9 cm. Calculate the angle between line GV and the plane of DEFG. V
F
D
12 cm
C
5 cm
O
Q
h) The diagram shows a pyramid with a triangle base CHD. Calculate the angle between line CA and the plane of ADH. A 8 cm B 6 cm
G
5 cm
D 2 cm H
E
9. 2 Angle Between Two Planes
Lines and Planes in 3-Dimensions
4
9.2.1
a) Calculate the angle between between the two planes. planes.
Example 1: Plane EFGH and plane GHDA
1. a) Plane KLSP and plane JKLM S
A
W
R
D
9 cm B
P
6 cm 8 cm
L
M
E
V
Q
C F
G
b) Plane PSWV and plane VUXW
15 cm
X
6 cm
S
12 cm
H
J
20 cm
4 cm
U
R
5 cm
K
P
7 cm
Q
Angle : ∠ DHE = ∠ AGF
tan ∠DHE = =
AF GF 9
6 DHE = 56.3 56.31 1o / 56o19’ ∠DHE Example 2 : Plane PQLK and plane SRLK
2. a) Plane ABCD and plane ADEF
b) Plane URST URST and plane XRSY
E K
L
13 cm
7 cm S P
F
D
R 10 cm
A
Angle : ∠ QLR = ∠ PKS
tan ∠QLR = =
10 cm
Y
C 5 cm 20 cm
Q
12 cm
X
T
U 9 cm
B R
S
12 cm
QR LR 10 7
o ∠QLR = 55
Example 3 : Plane TRQ and plane SRQP 11 cm P
3. a) Plane ABCD and plane V ABV Q
b) Plane PQSR and plane plane PQKL K L 3 cm
D SPlanes in 3-Dimensions Lines and
R A
T
C
5
B
S P
5 cm
R 4 cm Q
8 cm
5 cm 4 cm
5 cm
Angle : ∠ TRS TS
tan ∠TRS = =
RS 4 11
QLR = 19.9 19.98 8o / 19o59’ ∠QLR
Example 4: Plane DEV and DEFG . VO = 7 cm
4a) Plane GCB and plane ABCD G
V
G
F 12 cm
O
D
M
10 cm
E
L12 cm
8 cm
=
MO
7 6
o o ∠VMO = 49.40 / 49 24’
Lines and Planes in 3-Dimensions
L
F
6
M 10 cm
P
VO
N
T
B
Angle : ∠ VMO
tan ∠VMO =
12 cm
9 cm
C
O A
K
10 cm
D
b) Plane PMNT and Plane KLMN
Example 5 : Plane ABE and plane ABCD
4 a) Plane SRQ and plane SRUT
E
b) Plane SURP and plane PTR
P
F
D
K
C
L 18 cm
S
15
T
2
10 cm
−9
tan ∠ELK = =
U
W
5 cm S
B
U
N
R
Angle : ∠ ELK
EK =
4 cm Q
Q
36 cm A
V
P
M
R
12 cm
15 cm
T
8 cm
= 12 EK
2
LK 12
36
∠ELK =
Exercise 1
a) The diagram shows shows a pyramid pyramid with a horizontal base ABCD. Given that VO = 9 cm. Calculate the angle between the plane VAD and the plane of ABCD. V
b) The diagram shows a cuboid with with a horizontal base JKLM .Calculate the angle between the plane SRKJ and the plane of SRLM. R S
10 cm P A
L
M
8 cm
O B
Q
D
9 cm
6 cm C J
K
7 cm
C 10 cm
D shows S d) The diagram a prism. R the B Calculate
c) The diagram shows a prism. Calculate the
15 cm Lines and Planes in 3-Dimensions
7
P
A
8 cm
Q
angle between the the plane PLM and the plane of PLNQ. P
angle between the the plane QRC and the plane of of PQRS.
Q
5 cm N
L 10 cm K
20 cm
M
e) The diagram shows shows a pyramid. pyramid. Calculate f) The diagram shows shows a prism. Name the the angle the angle between between the plane FGP FGP and the plane between the plane ABCD and the plane of of EFGH DQR. P D
C 9cm
14cm
A
H
S
G
18 cm
P
F
How to answer answer the SPM format Question
Lines and Planes in 3-Dimensions
R 7cm
24 cm E
B
8
13 cm
Q
Step 3 : Identify the angle
Example 1 Diagram 1 shows a pyramid LPQRS .
L
L
P P
S
J cm
S 10 cm
J cm
10 cm
Q Q
12 cm
R
12 cm
Angle :
∠ LQJ
Diagram 1 The base PQRS is a horizontal rectangle. J is the midpoint of RS. The vertex L is 8 cm vertically above the point J. Calculate the angle between the line QL and the base PQRS.
Step 4 : Calculate the angle
Step 1 : - Colour line QL and shade/colour plane PQRS - Determine the meet point L
JQ =
12
2
−5
tan ∠LQJ = P
S
J cm
10 cm Q
= ∠LQJ =
R
12 cm
Step 2 : Identify normal and orthogonal projection L
P
S
J cm
10 cm Q
12 cm
R
Normal line : LJ Orthogonal projection : QJ
Lines and Planes in 3-Dimensions
9
2
= 13
LJ QJ
8 13
R
Step 3 : Identify the perpendicular line with BC and lies on plane NBC and the base ABCD .
Example 2 Diagram 2 shows a prism with horizontal square ABCD. Trapezium KABL is the uniform cross-section of the prism. The rectangular surface NKAD is vertical while the rectangular surface MLBC is inclined.
N
N
M
L
M
K
C
D 6 cm
L K
C
D
A
6 cm
A
Diagram 2
Step 4 : Identify the angle N
Calculate the angle between the plane NBC and the base ABCD. Step 1 : - Shade/colour Shade/colour plane ABCD - Determine the line intersection between plane NBC and the base ABCD N
M
L K
C
D 6 cm
M A
8 cm
L K
C
D
Angle :
∠ NCD
6 cm
A
B
Line NC and DC are perpendicular with line BC
B
8 cm
8 cm
8 cm
Step 5 : Calculate the angle ND ND tan ∠ NCD = DC
B
Line intersect : BC = ∠ NCD
Questions Based on the Examination Format
Lines and Planes in 3-Dimensions
10
6 8 o = 36.89 / 36o52’
B
1. Diagram 1 shows shows a pyramid pyramid with a 2. Diagram 2 shows a cuboid with horizontal rectangular base PQRS. V is vertically above P. base KLMN. S
V
R 4 cm
P 11 cm
N
S
M
R
5 cm
6 cm P
Q
K
Q
8 cm
L
12 cm DIAGRAM 2
DIAGRAM 1 Calculate the angle between the line VR and the plane PQRS.
Calculate the angle between the line SL and the base NKLM.
3. Diagram 3 shows a cuboid ACBDEFGH. Given EH = FG = 8 cm.
4. Diagram 4 shows a right prism with a horizontal plane ABCD. It is a uniform prism and its cross section is an isosceles triangle of sides 4 cm. The thickness of the prism, EA = 4 cm. H
D
C 7 cm B
A
6 cm
H
E
G
F
5 cm
A
E
DIAGRAM 3 Calculate the angle between the plane EHD and the plane FEHG.
C
D
4 cm
B
DIAGRAM 4 Calculate the angle between the plane ABH and the plane ABE.
S
T
4 cm 5) Diagram 5 shows a pyramid with the
Z 6) R Diagram 6 shows a cuboid. Z is the U 6 cm
Lines and Planes in 3-Dimensions
X
W
11
Y
V6 10DIAGRAM cm
horizontal plane, TRS. The rectangle PQRS is vertical plane. 10 cm
P
midpoint of TW .
Q
12 cm 13 cm
S
R
T DIAGRAM 5 Calculate the angle between the plane PTS and the plane TQR.
7) Diagram 7 shows a right prism with with base the rectangular plane ABCD. Right triangle BCF is the uniform cross-section of the prism. The rectangular surface DCFE is vertical while the rectangular surface BAEF is is inclined. E
Calculate the angle between plane YVZ and the horizontal plane XYVW.
8) Diagram 8 shows a pyramid REFGH. The The base EFGH is a horizontal rectangle. rectangle. R is the midpoint of HG. The apex R is 9 cm vertically above the point S. R
F H
6 cm D
•
S
G
5 cm
C
6 cm A
8 cm
B
E
DIAGRAM 7
24 cm
F
DIAGRAM 8
Calculate the angle between the plane DB and plane EDCF.
Calculate the angle between line ER and the plane EFGH.
S R
Y
K N
P a cuboid. 9) Diagram Diag cuboid. P is the thQe midpoint midpoint 12ram cm 9 shows
10) Diagram 10 shows shows a right prism. Right Right
10 cm Lines and Planes in 3-Dimensions
12
M L
6 cm
of line RQ.
angled triangle SUT is the uniform crosssection of the prism. P 5 cm
12 cm
R
Q
20 cm U T
S
DIAGRAM 10
DIAGRAM 9 cm Calculate the angle between the plane LQY and the plane MQRN.
Calvulate the angle between the plane PSR and the plane PUTR..
11) Diagram 11 shows shows a prism . The base base PQRS is a horizontal rectangle . X is the midpoint of SR.
12) Diagram 12 shows shows a right prism prism with rectangle base EFGH. EFPQ and GHPQ are rectangle. P
L
M Q
5 cm E S P
X
12 cm
R 8 cm
F
DIAGRAM 11 cm
M
G
6 cm
DIAGRAM 12 cm Calculate the angle between line LQ and the base EFGH.
Past Year SPM Questions
Lines and Planes in 3-Dimensions
5 cm
H
12 cm
Q
Calculate the angle between line PX and the plane SRML.
L
13
1. Nov 2003
Diagram 1 shows a prism with a horizontal square base HJKL. Trapezium EFLK is the uniform cross-section of the prism. The rectangular surface DEKJ is vertical while the rectangular surface GFLH is incline. D
G
F
E
H J
6 cm
K
L
8 cm Diagram 1
Calculate the angle be between the plane DL DLH and the base HJKL.
[ 4 marks ]
2 July 2004, Q4
Diagram 2 shows a cuboid. E
H
D
C
9 cm
F
G
5 cm DIAGRAM 2 A
12 cm
B
Calculate the angle between the line AH and the plane ABCD.
Lines and Planes in 3-Dimensions
14
[4 marks]
3. Nov 2004, Q3
Diagram 2 shows a pyramid VJKLM. V
L
M
Q cm
10 cm K
J
12 cm
DIAGRAM 2 The base JKLM is a horizontal horizontal rectangle. Q is the midpoint of JM. The apex V is 8 cm vertically above the point Q. Calculate the angle betw etween the line KV and the base JKLM.
[ 4 marks rks ]
4. July 2005, Q2
Diagram 1 shows a right prism with rectangle ABCD as its horizontal base. Right angled triangle FAB is the uniform cross-section of the prism. The rectangular surface BCEF is inclined. E
F D
C
3 cm 12 cm A
5 cm
B DIAGRAM 1
Calculate Calculate the angle angle between between the plane plane ABE and the base ABCD. ABCD.
Lines and Planes in 3-Dimensions
15
[3 marks] marks]
5. Nov 2005, Q4
Diagram 1 shows a right prism. Right angled triangled PQR is the uniform cross-section of the prism. 12 cm
T
S
5 cm U
18 cm Q
R
P
DIAGRAM 1
Calculate the angle between the plane RTU and the plane PQTU.
6. July 2006, Q4
Diagram 2 shows a right prism. The base HJKL is a horizontal rectangle. The right angled triangle NHJ is the uniform cross-section of the prism. M
N
8 cm L K 6 cm H 12 cm
J
DIAGRAM 2 Identify and calculate the angle between the line KN and the plane HLMN. 7. Nov 2006, Q2
Lines and Planes in 3-Dimensions
16
Diagram 1 shows a right prism. The base PQRS is on horizontal rectangle. The right triangle UPQ is the uniform cross section of the prism.
Identify and calculate the angle between the line RU and the base PQRS. [ 4 marks ]
8. SPM June 2007 Q2
Diagram shows a right prism. The base PQRS is a horizontal rectangle. Trapezium PQVU is the uniform cross-section of the prism. The rectangle QRWV is a vertical plane and the rectangle UVWT is an T inclined plane. U
W
14 cm
V
S P
7 cm
R 12 cm
5 cm
Q
Identify and calculate the angle between the plane PQW and the base PQRS. [3 marks]
Lines and Planes in 3-Dimensions
17
9. SPM SPM Nov 2007 2007 Q4
Diagram shows a right prism. The base PQRS is a horizontal rectangle. Right angled triangle QRU is the uniform cross-section of the prism. V is the midpoint of PS. T
U S 5 cm R
V
16 cm
P 12 cm Q
Identify and calculate the angle between the line UV and the plane RSTU. [3 marks]
10. SPM June June 2008 2008
Diagram shows a cuboid ABCDEFGH with horizontal base ABCD. P, Q and R are the midpoints of BC, AD and FE respectively. E R
H
F G 5 cm D Q C A
P
8 cm
B
Name and calculate the angle between the plane
Lines and Planes in 3-Dimensions
6 cm
18
FPCR and the base ABCD. [4 marks]
11. SPM Nov 2008
Diagram shows a cuboid. M is the midpoint of the side EH and AM = 15 cm.
E M F
H G
D
A
C 8 cm B
a) Name the angle angle between between the line line AM and the plane plane ADEF. ADEF. b) Calculate Calculate the the angle between between the the line line AM and the the plane ADEF. ADEF. [3 marks]
Lines and Planes in 3-Dimensions
19
ANSWERS Chapter 9 :Lines And Planes In 3 Dimensions Dimensions
9.1.1 1a
16.70o / 16o42’
1b
54.46 o / 54o28’
2a
68.20o / 68o12’
2b
3a
21.80o / 21o48’
3b
24.78 o / 24o47’
3c
28.30o / 28o18’
3d
3e
18.43o / 18o26’
b
26.57o / 26o34’
c
54.46o / 54o28’
d
f
51.34o / 51o20’
g
54.16o / 54o10’
h
1b
74.05o / 74o3’
2a
67.38o / 67o23’
2b
3b
36.89o / 36o52’
4a
60o
4b
5b
63.43o / 63o26’
b
33.69o / 33o41’
f
34.70o / 34o42’
Exercise 1 a 50.91o / 50o54’ e 28.30o / 28o18’ 9.2.1 1a 36.87o / 36o52’ 3a 57.99o / 58o 5a
36.87o / 36o52’
Exercise 1 a 66.04o / 66o2’ e
37.87o / 37o52’
29.74 o / 29o45’ 38.66 o / 38o40’
71.57 o / 71o34 53.13 o / 53o8’ 29.05 o / 29o3’ 53.13o / 53o8’
c
26.57o / 26o34’
d
66.42o / 66o25’
54.46o / 54o28’ 36.87 o / 36o52’ 53.13 o / 53o8’
4
56.31o / 56o19’ 34.70o / 34o42’ 18.43o / 18o26’
PRACTICE SPM FORMAT 1 5 9
47.73o / 47o44’ 63.43o / 63o26’ 30.96o / 30o58’
Lines and Planes in 3-Dimensions
2
17.10o / 17o6’
3
6
36.87o / 36o52’
7
10
30.96o / 30o58’
11
20
8 12
SPM PAST YEAR QUESTIONS
1 2 3 4 5 6 7 8 9 10
Nov 2003 Jul 2004 Nov 2004 Jul 2005 Nov 2005 Jul 2006 Nov 2006 Jun 2007 Nov 2007 Jun 2008
