Chapter 9 II Lines & Planes in 3D ENHANCE
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Transcript of Chapter 9 II Lines & Planes in 3D ENHANCE
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8/3/2019 Chapter 9 II Lines & Planes in 3D ENHANCE
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CHAPTER 9 : LINES AND PLANES IN 3 DIMENSIONS
9. 1 Angle Between Lines And Planes
9.1.1 a)Based on the diagram, calculate the angle between the line and the plane given
Example 1: Plane :EFGH
Line :GC
Angle : CGH
tan CGH =GH
CH
=8
4
CGH = 26.57o / 26o 34
1. a) Plane : ABCD
Line : DV
Angle :
b) Plane : SRLK
Line : QL
Angle :
Example 2 : Plane : PSK
Line : KR
Angle : RKS
tan RKS =KS
SR
=5
12
RKS = 67.38o / 67o 23
2. a) Plane : CDEH
Line : FD
Angle :
b) Plane : URST
Line : RX
Angle :
Example 3 : Plane : JKLM 3. a) Plane : ABCD b) Plane : ABCD
Lines and Planes in 3-Dimensions 1
PQ
RS
LK
G H
EF
DA
B C
6 cm
8 cm
4 cm
PQ
RS
LK
12 cm
7 cm
5 cm
AB
CD
V
10 cm
8 cm
3 cm
12 cm
5 cm
GH
EF
DA
B C
15 cm
6 cm
8 cm
R S
UT
YX
24 cm
7 cm
4 cm
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Line : NK
NM = 11 cm
Angle : NKM
KM = 22 912 + = 15 cm
tan NKM =KM
NM
=15
11
NKM = 36.25o / 36o 15
Line : AV
Line : DG
c) Plane : QRSTLine : TP
d) Plane : QPWTLine : RX
e) Plane : SRUTLine : PN
Exercise 1 : Based on the diagram, calculate the angle between the line and the plane given
Lines and Planes in 3-Dimensions 2
J K
LM
N
12 cm
9 cm
AB
CD
V
8 cm
6 cm
4 cm
G H
EF
DA
B C
6 cm
5 cm
12 cm
5 cm R
T
Q
P
S12 cm
7 cm
Q P
WS
VU
R T
X
Y
8 cm
12 cmR
U
S
P
T
Q
N
M
6 cm
5 cm
12 cm
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a) The diagram shows a cuboid. Calculate the
angle between line NE and the plane of GFKN
b) The diagram shows a cuboid with a
horizontal base JKLM .Calculate the angle
between line KS and the plane of SRLM.
c) The diagram shows a prism. Calculate the
angle between line RY and the plane of STY.
d) The diagram shows a prism. Calculate the
angle between line QE and the plane of DCE.
e) The diagram shows a pyramid . Given that
HP = 13 cm. Calculate the angle between line
f) The diagram shows a prism. Calculate the
angle between line UV and the plane of PSWV.
Lines and Planes in 3-Dimensions 3
J K
Q
M
RS
P
L
6 cm
5 cm
8 cm
R S
UT
YX
6 cm
8 cm
14 cm
BA
F
E
D C
Q
P
6 cm
5 cm
12 cm
GH
EF
LK
N M
16 cm
5 cm
12 cm
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PG and the plane of EHP.
g) The diagram shows a pyramid with a
horizontal base DEFG. Given that VO = 9 cm.
Calculate the angle between line GV and the
plane of DEFG.
h) The diagram shows a pyramid with a
triangle base CHD. Calculate the angle
between line CA and the plane of ADH.
9. 2 Angle Between Two Planes
Lines and Planes in 3-Dimensions 4
E F
GH
P
7 cm
9 cmP Q
RS
X
W
V
U
5 cm
4 cm
3 cm7 cm
DE
FG
V
O
12 cm
5 cm
D
B
C
H
A
6 cm
8 cm
2 cm
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9.2.1 a) Calculate the angle between the two planes.
Example 1: Plane EFGH andplane GHDA
Angle :
DHE = AGF
tan DHE =GF
AF
=6
9
DHE = 56.31o / 56o19
1. a) Plane KLSP and plane
JKLM
b) Plane PSWV and plane
VUXW
Example 2 : Plane PQLK and
plane SRLK
Angle :
QLR = PKS
tan QLR =LR
QR
=710
QLR = 55o
2. a) Plane ABCD and plane
ADEF
b) Plane URST and plane
XRSY
Example 3 : Plane TRQ and
plane SRQP
3. a) Plane ABCD and plane
ABV
b) Plane PQSR and plane
PQKL
Lines and Planes in 3-Dimensions 5
AB
CD
V
GH
EF
DA
B C
PQ
RS
LK
T
RS
QP
8 cm
6 cm
9 cm
P Q
RS
X
W
V
U
7 cm
4 cm
6 cm
5 cmJ K
Q
M
RS
P
L
20 cm
12 cm
15 cm
PQ
RS
LK
12 cm
10 cm
7 cm
BA
F
E
D C
20 cm
10 cm
13 cm
5 cm
11 cm
4 cm
3 cm
R S
UT
YX
12
cm
9 cm
5 cm
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Angle : TRS
tan TRS =RS
TS
=11
4
QLR = 19.98o / 19o59
Example 4: Plane DEV and
DEFG . VO = 7 cm
Angle : VMO
tan VMO =MO
VO
=6
7
VMO = 49.40o / 49o24
4a) Plane GCB and plane
ABCD
b) Plane PMNT and Plane
KLMN
Lines and Planes in 3-Dimensions 6
AB
CD
G
OL
5 cm
4 cm
8 cm
5 cm
DE
FG
V
OM
10 cm
12 cm
8 cm
12 cm
10 cm
T
L M
NK
F
P
9 cm
12 cm
10 cm
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Example 5 : Plane ABE and
plane ABCD
Angle : ELK
EK = 22 915 = 12
tan ELK =LK
EK
=3612
ELK =
4 a) Plane SRQ and plane
SRUT
b) Plane SURP and plane PTR
Exercise 1
a) The diagram shows a pyramid with a
horizontal base ABCD. Given that VO = 9 cm.
Calculate the angle between the plane VAD
and the plane of ABCD.
b) The diagram shows a cuboid with a
horizontal base JKLM .Calculate the angle
between the plane SRKJ and the plane of
SRLM.
c) The diagram shows a prism. Calculate the d) The diagram shows a prism. Calculate the
Lines and Planes in 3-Dimensions 7
BA
F
E
D C
L
K
18 cm
15 cm
36 cm
R
U
S
P
T
Q
N
M
8 cm
5 cm
Q
P
WSU
R
T
V
10 cm
4 cm
12 cm
J K
Q
M
RS
P
L
BC
DA
V
O
10 cm
8 cm
7 cm
6 cm9 cm
QP
D
C
S R
A
B
8 cm
10 cm
15 cm
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angle between the plane PLM and the plane of
PLNQ.
angle between the plane QRC and the plane of
PQRS.
e) The diagram shows a pyramid. Calculate
the angle between the plane FGP and the plane
of EFGH
f) The diagram shows a prism. Name the angle
between the plane ABCD and the plane of
DQR.
How to answer the SPM format Question
Lines and Planes in 3-Dimensions 8
K M
LN
QP
20 cm
10 cm
5 cm
E F
GH
P
18 cm
24 cm
14cm
PQ
RS
CD
A B
13 cm
7cm
9cm
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Example 1
Diagram 1 shows a pyramid LPQRS .
The base PQRS is a horizontal rectangle. J is
the midpoint of RS. The vertex L is 8 cmvertically above the point J. Calculate the angle
between the line QL and the base PQRS.
Step 1 :
- Colour line QL and shade/colour plane PQRS
- Determine the meet point
Step 2 :
Identify normal and orthogonal projection
Normal line : LJ
Orthogonal projection : QJ
Step 3 :
Identify the angle
Angle : LQJ
Step 4 :
Calculate the angle
JQ = 22 512 = 13
tan LQJ =QJ
LJ
=
13
8
LQJ =
Lines and Planes in 3-Dimensions 9
QR
SP
L
10 cm
12 cm
Diagram 1
J
cm
Q R
SP
L
10 cm
12 cm
J
cm
Q R
SP
L
10 cm
12 cm
J
cm
Q R
SP
L
10 cm
12 cm
J
cm
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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.
Calculate the angle between the plane NBC and
the base ABCD.
Step 1 :
- Shade/colour plane ABCD
- Determine the line intersection between plane
NBC and the base ABCD
Line intersect : BC
Step 3 :
Identify the perpendicular line with BC and lies
on plane NBC and the base ABCD .
Line NC and DC are perpendicular with line
BC
Step 4 : Identify the angle
Angle : NCD
Step 5 :
Calculate the angle
tan NCD =DC
ND
=8
6
NCD = 36.89o / 36o52
Questions Based on the Examination Format
Lines and Planes in 3-Dimensions 10
A
L
N M
CD
K
B
6 cm
8 cm
Diagram 2
A
L
N M
CD
K
B
6 cm
8 cm
A
L
N M
CD
K
B
6 cm
8 cm
A
L
N M
CD
K
B
6 cm
8 cm
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1. Diagram 1 shows a pyramid with a
rectangular base PQRS. V is vertically above P.
Calculate the angle between the line VR and
the plane PQRS.
2. Diagram 2 shows a cuboid with horizontal
base KLMN.
Calculate the angle between the line SL and the
base NKLM.
3. Diagram 3 shows a cuboid ACBDEFGH.
Given EH = FG = 8 cm.
Calculate the angle between the plane EHD and
the plane FEHG.
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.
Calculate the angle between the plane ABH and
the plane ABE.
5) Diagram 5 shows a pyramid with the 6) Diagram 6 shows a cuboid. Z is the
Lines and Planes in 3-Dimensions 11
DIAGRAM 1
DIAGRAM 2
DIAGRAM 3
A B
C
H
E D
DIAGRAM 4
KL
MN
RS
P Q
12 cm
4 cm
5 cm
P Q
RS
V
8 cm
6 cm
11 cm
F E
HB
CD
A
G
7 cm
5 cm
6 cm
4 cm
YV
WX
TS
R U
10 cm
6 cm
4 cm
Z
DIAGRAM 6
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horizontal plane, TRS. The rectangle PQRS is
vertical plane.
Calculate the angle between the plane PTS and
the plane TQR.
midpoint of TW .
Calculate the angle between plane YVZ and the
horizontal plane XYVW.
7) Diagram 7 shows a right prism 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 inclined.
Calculate the angle between the plane DB and
plane EDCF.
8) Diagram 8 shows a pyramid REFGH. The
base EFGH is a horizontal rectangle. R is the
midpoint of HG. The apex R is 9 cm vertically
above the point S.
Calculate the angle between line ER and the
plane EFGH.
9) Diagram 9 shows a cuboid. P is the midpoint 10) Diagram 10 shows a right prism. Right
Lines and Planes in 3-Dimensions 12
T
RS
QP
12 cm
13 cm
10 cm
DIAGRAM 5
B
DIAGRAM 7 DIAGRAM 8
EF
GH
R
5 cm
24 cm
S
LM
QP
RS
KN
Y
10 cm
6 cm
12 cm
A
CD
FE
8 cm
6 cm
6 cm
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of line RQ.
Calculate the angle between the plane LQY and
the plane MQRN.
angled triangle SUT is the uniform cross-
section of the prism.
Calvulate the angle between the plane PSR and
the plane PUTR..
11) Diagram 11 shows a prism . The base
PQRS is a horizontal rectangle . X is the
midpoint of SR.
Calculate the angle between line PX and the
plane SRML.
12) Diagram 12 shows a right prism with
rectangle base EFGH. EFPQ and GHPQ are
rectangle.
Calculate the angle between line LQ and the
base EFGH.
Past Year SPM Questions
Lines and Planes in 3-Dimensions 13
DIAGRAM 9 cm
U
Q
ST
P
R
5 cm12 cm
20 cm
DIAGRAM 10
PQ
RS
ML
X
12 cm
8 cm
5 cm
DIAGRAM 11 cm
F
G
E
P
H
Q
M
L
6 cm
5 cm
12 cm
DIAGRAM 12 cm
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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 surfaceGFLH is incline.
Calculate the angle between the plane DLH and the base HJKL. [ 4 marks ]
2 July 2004, Q4
Diagram 2 shows a cuboid.
Calculate the angle between the line AH and the plane ABCD. [4 marks]
Lines and Planes in 3-Dimensions 14
K
F
D G
H
J
E
L
6 cm
8 cm
Diagram 1
A B
G
D C
E
F
H
12 cm
5 cm
9 cm
DIAGRAM 2
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3. Nov 2004, Q3
Diagram 2 shows a pyramid VJKLM.
The base JKLM is a horizontal rectangle. Q is the midpoint of JM. The apex V is 8 cmvertically above the point Q.
Calculate the angle between the line KV and the base JKLM. [ 4 marks ]
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.
Calculate the angle between the plane ABE and the base ABCD. [3 marks]
Lines and Planes in 3-Dimensions 15
B
E
A
CD
F
12 cm
5 cm
3 cm
DIAGRAM 1
DIAGRAM 2
K J
ML
V
10 cm
12 cm
Q
cm
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5. Nov 2005, Q4
Diagram 1 shows a right prism. Right angled triangled PQR is the uniform cross-section of
the prism.
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.
Identify and calculate the angle between the line KN and the plane HLMN.
7. Nov 2006, Q2
Lines and Planes in 3-Dimensions 16
U
Q
ST
P
R
12 cm
5 cm
18 cm
DIAGRAM 1
DIAGRAM 2
J
M
H
KL
N
6 cm
12 cm
8 cm
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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. TrapeziumPQVU is the uniform cross-section of the prism. The rectangle QRWV is a vertical plane
and the rectangle UVWT is an
inclined plane.
Identify and calculate the angle between the plane PQW and the base PQRS.
[3 marks]
Lines and Planes in 3-Dimensions 17
P R
W
Q
12 cm
T
S
U
7 cm
14 cm
5 cm
V
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9. SPM Nov 2007 Q4
Diagram shows a right prism. The base PQRS is a horizontal rectangle. Right angledtriangle QRU is the uniform cross-section of the prism. V is the midpoint of PS.
Identify and calculate the angle between the line UV and the plane RSTU.[3 marks]
10. SPM June 2008
Diagram shows a cuboid ABCDEFGH with horizontal base ABCD. P, Q and R are themidpoints of BC, AD and FE respectively.
Name and calculate the angle between the plane FPCR and the base ABCD.
[4 marks]
Lines and Planes in 3-Dimensions 18
P
Q
R
S
T
U
V
16 cm
12 cm
5 cm
A
B
C
D
E
F
G
H
P
R
Q
6 cm8 cm
5 cm
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11. SPM Nov 2008
Diagram shows a cuboid. M is the midpoint of the side EH and AM = 15 cm.
a) Name the angle between the line AM and the plane ADEF.
b) Calculate the angle between the line AM and the plane ADEF.[3 marks]
Lines and Planes in 3-Dimensions 19
A
B
C
D
E
F
G
H
M
8 cm
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ANSWERSChapter 9 :Lines And Planes In 3 Dimensions
9.1.1
1a 16.70o / 16o42 1b 54.46o / 54o28 2a 68.20o / 68o12 2b 29.74o /
29o45
3a 21.80o / 21o48 3b 24.78o / 24o47 3c 28.30o / 28o18 3d 38.66o /
38o40
3e 18.43o / 18o26
Exercise 1
a 50.91o /50o54
b 26.57o / 26o34 c 54.46o / 54o28 d 71.57o /71o34
e 28.30o /
28o18
f 51.34o / 51o20 g 54.16o / 54o10 h 53.13o /
53o8
9.2.1
1a 36.87o /
36o52
1b 74.05o / 74o3 2a 67.38o / 67o23 2b 29.05o /
29o3
3a 57.99o / 58o 3b 36.89o / 36o52 4a 60o 4b 53.13o /
53o8
5a 36.87o /
36o52
5b 63.43o / 63o26
Exercise 1
a 66.04o / 66o2 b 33.69o / 33o41 c 26.57o /
26o34
d 66.42o /
66o25
e 37.87o /
37o52
f 34.70o / 34o42
PRACTICE SPM FORMAT
1 47.73o /
47o44
2 17.10o / 17o6 3 54.46o /
54o28
4 56.31o /
56o19
5 63.43o /
63o26
6 36.87o / 36o52 7 36.87o /
36o52
8 34.70o /
34o42
9 30.96o /
30o58
10 30.96o / 30o58 11 53.13o /
53o8
12 18.43o /
18o26
Lines and Planes in 3-Dimensions 20
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SPM PAST YEAR QUESTIONS
1 Nov 2003 36.87o / 36o 52
2 Jul 2004 18.43o / 18o 26
3 Nov 2004 31.61o
/ 31
o
364 Jul 2005 14.04o / 14o 2
5 Nov 2005 33.69o / 33o 41
6 Jul 2006 50.19o / 50o 12
7 Nov 2006 34.70 / 34O42
8 Jun 2007 ,54.46 or 54 28'WQR
9 Nov 2007 SUV , 31.61 or 31 36'
10 Jun 2008 ,32QPR
11 Nov 2008 ,15.47 or 15 28'EAM