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Line And Planes In 3-Dimensions 1
CHAPTER 17 : LINES AND PLANES IN 3 DIMENSIONS
17.1 Angle Between Lines And Planes
Definitions
17.1 The angle between a line and a plane is the angle between the line and its orthogonal
projection on the plane.
1. 2.
O
C
AB
D
E
C
D
BA
E
Line DE and plane ABCD
Line AE and plane ABCD
3. 4.
CD
BA
E
CD
BA
E
Line CE and plane ABCD
Line BE and plane ABCD
P Q
RS
K
L
V
o
Line: KV
Plane : PQRSNormal to the plane: VL
Orthogonal projection of the lineonto the plane: KL
Angle between KV and the plane
PQRS is VKL
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Line And Planes In 3-Dimensions 2
17.1.1 Name and draw the i) normal line
ii) orthogonal projection
Example 1: Plane :EFGH
Line :GD
Normal line : DE
Orthogonal projection : GE
1. a) Plane : DEFGLine : DV
Normal line :
Orthogonal projection :
b) Plane : PQRSLine : KQ
Normal line :
Orthogonal projection :
Example 2 : Plane : PSK
Line : KR
Normal line : SROrthogonal projection : SK
2. a) Plane : CDEH
Line : GC
Normal line :Orthogonal projection :
b) Plane : BCV
Line : AV
Normal line :Orthogonal projection :
Example 3 : Plane : GEVLine : VF
Normal line : FOOrthogonal projection : VO
3. a) Plane : CDVLine : BV
Normal line :Orthogonal projection :
b) Plane : ADEFLine : DG
Normal line :Orthogonal projection :
G H
EF
DA
B C
G H
EF
DA
B C
P Q
RS
LK
ED
FG
V
O A B
CD
V
A B
CD
V
G H
EF
DA
B C
DE
FG
V
O
P Q
RS
LK
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Line And Planes In 3-Dimensions 3
17.1.2 (a) Name the angle between the line and the plane given
Example 1: Plane :EFGH
Line :GD
Angle : DGE
1. a) Plane : DEFG
Line : DV
Angle :
b) Plane : PQRS
Line : KQ
Angle :
Example 2 : Plane : PSKLine : KR
Angle : RKS
2. a) Plane : CDEHLine : GC
Angle :
b) Plane : BCVLine : AV
Angle :
Example 3 : Plane : GEV
Line : VF
Angle : FVO
3. a) Plane : CDV
Line : BV
Angle :
b) Plane : ADEF
Line : DG
Angle :
G H
EF
DA
B C
G H
EF
DA
B C
P Q
RS
LK
ED
FG
V
OA B
CD
V
A B
CD
V
G H
EF
DA
B C
DE
FG
V
O
P Q
RS
LK
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Line And Planes In 3-Dimensions 4
Exercise 1 : Name the angle between the line and the plane given
a) Line UN and plane PQU b) Line JN and plane JKLM
c) Line XS and plane XYTU d) Line BE and plane ABCD
e) Line MF and plane KLMN f) Line PA and plane PQRS
J K
Q
M
RS
P
L
yN
y G
T
P Q
RS yN
y M
U
RS
U T
YX
BA
F
E
D C
G H
EF
LK
N M
P Q
RS
X
W
V
Ay
By
y L
y K
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Line And Planes In 3-Dimensions 5
Exercise 2 : Name the angle between the line and the plane given
a) The diagram shows a pyramid with a
horizontal base DEFG. Name the angle
between line GV and the plane of DEFG.
b) The diagram shows a cuboid with a
horizontal base JKLM .Name the angle
between line KS and the plane of SRLM.
c) The diagram shows a prism. Name theangle between line RY and the plane of STY.
d) The diagram shows a prism. Name theangle between line QE and the plane of
ABCD.
e) The diagram shows a cuboid. Name theangle between line NE and the plane of GFKN
f) The diagram shows a prism. Name the anglebetween line RV and the plane of PSWV.
J K
M
RS
P
L
R
S
U T
YX
BA
F
E
D C
G H
EF
LK
N M
P Q
RS
X
W
V
U
y Q
y P
DE
FG
V
O
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Line And Planes In 3-Dimensions 6
17.2 Angle Between Two Planes17.2 The angle between 2 intersecting planes is the angle between 2 lines; one on each plane,which are drawn respectively from a common point on the line of intersection between the 2
planes and perpendicular to it.
Skills assessed
y To identify the angle between a line and a given plane.
y To identify the angle between 2 given planes.
Solving Strategies :
y Draw and/or colour the given line.
y Shade or colour the given plane.
y Draw the normal to the plane.
y Draw the orthogonal projection of the line onto the plane.
y Mark the angle between the line and its orthogonal projection onto the plane.
y Name the angle using the 3 alphabets.
Common errors :
y Students failed to identify the given plane.
y Students did not find the orthogonal projection of the line onto the plane.
y Student failed to identify the required angle.
LM and MN are perpendicular toDC.
The angle between the planeABCD and the plane CDEF is
KML ( or ADE or BCF )
M
L
NAB
CD
E F
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Line And Planes In 3-Dimensions 7
Finding Normal and Orthogonal Projection
1. 2.
Plane ABCD and plane CDEF
E
C
AB
D
F
PlaneABCD and plane CDEF
E
CD
BA
F
3. 4.
Plane ABCD and plane CDE
F
C
AB
E
PlaneABCD and plane CDG
J
H
G
F
CD
BA
E
5. 6.
Plane ABCD and plane ADE
F
C
AB
D
E
Plane ABCD and plane ADH
H
G
F
CD
BA
E
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Line And Planes In 3-Dimensions 8
17.2.1 a) Name the angle between the two planes.
Example 1: Plane EFGH andplane GHDA
Angle :
DHE and AGF
1. a) Plane KLSP and plane
JKLM
b) Plane PSWV and plane
VUXW
Example 2 : Plane PQLK and
plane SRLK
Angle : QLR and PKS
2. a) Plane ABCD and plane
ADEF
b) Plane URST and plane
XRSY
Example 3 : Plane TRQ and
plane SRQP
Angle : TRS
3. a) Plane ABCD and plane
ABV
b) Plane PQSR and plane
PQK
A B
CD
V
J K
Q
M
RS
P
L
P Q
R
X
W
U
G H
EF
DA
B C
BA
F
E
D C
RS
U T
YX
P Q
RS
LK
P Q
RS
LK
T
R
QP
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Line And Planes In 3-Dimensions 9
c) Plane JKLM and planePKL
d) Plane MHEL and planeNHE
e) Plane ABCD and planeBCE
Example 4: Plane DEV andDEFG
Angle : VMO
4a) Plane GCB and planeABCD
b) Plane KLMN and planeKPN
c) Plane ABE and plane
ABCD
d) Plane RUQ and plane
SRUT
e) Plane SURP and plane PTR
J K
Q
M
RS
P
L
BA
F
E
D C
G H
EF
LK
N M
D E
FG
V
O
y M A B
CD
G
O y L
BA
F
E
D C
y L
y K
R
S
T
Q
yN
y M
T
LM
NK
y E
y F
P
Q
y W
R
T
y V
S
P
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Line And Planes In 3-Dimensions 10
Exercise 1 : Name the angle between the two planes
a) Plane SRQP and plane QUTR b) Plane JQRM and plane JKLM
c) Plane RSYX and plane URST d) Plane BCF and plane ABCD
e) Plane GMLF and plane GHEF f) Plane PQA and plane PQRS
Cy
J K
Q
M
RS
P
L
RS
U T
YX
BA
F
E
D C
G H
EF
LK
N M
P Q
RS
X
W
V
Ay
By
T
Q
R
U
P
S
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Line And Planes In 3-Dimensions 11
Exercise 2 : Name the angle between the two planes
a) The diagram shows a pyramid with a
horizontal base DEFG. Name the angle
between the plane VDE and the plane ofDEFG.
b) The diagram shows a cuboid with a
horizontal base JKLM .Name the angle
between the plane SRKJ and the plane ofSRLM.
c) The diagram shows a prism. Name theangle between the plane RSY and the plane of
RSTU.
d) The diagram shows a prism. Name theangle between the plane ABE and the plane of
ABCD.
e) The diagram shows a cuboid. Name theangle between the plane HEK and the plane of
GHEF
f) The diagram shows a prism. Name the anglebetween the plane UVWX and the plane of
PSWV.
J K
M
RS
P
L
RS
U T
YX
BA
F
E
D C
G H
EF
LK
NM
y Q
y P
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Line And Planes In 3-Dimensions 12
Questions According To SPM Format
1. Diagram 1 shows a pyramid with a
rectangular base PQRS. V is vertically above P.
Name the angle between the line VR and theplane PQRS
A. PRV B. VRS
C. PVQ D. VQS
2. Diagram 2 shows a cuboid.
Name the angle between the line PM and theplane SRMN.
A. PMK B. PMQ
C. PMR D. PMS
3. In the diagram 3, X and Y are the midpoints
of PW and SV respectively.
The angle between line RX and the plane
RSVU is
A. RXY B. XYR
C. XRY D. XQR
4. Diagram 4 shows a right prism.
Name the angle between line EK and plane
EFJH.
A. EKJ B. EKF
C. HEK D. JEK
K L
MN
RS
P Q
P Q
RS
V
DIAGRAM 1 DIAGRAM 2
Q P
WS
VU
R T
y X
DIAGRAM 3
y Y
F G
K
H
E J
DIAGRAM 4
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Line And Planes In 3-Dimensions 13
5. Diagram 5 shows a pyramid .PQRS is a
rectangle.
Name the angle between the line TQ and theplane PQRS
A. QPT B. QST
C. RST D. SQT
6. Diagram 6 shows a cuboid.
The angle between the line SU and planePSWT is
A. USP B. USQ
C. UST D. USW
7. Diagram 7 shows a cuboid.
The angle between plane QPV and the planeQPWT is
A. VQW B. UQT
C. VPW D. QPV
8. Diagram 8 shows a right prism.
Name the angle between the plane EGKH andthe plane FGKJ.
A. EGF B. EKF
C. HGJ D. GKE
T U
VW
RS
P Q
DIAGRAM 5
DIAGRAM 6
Q P
WS
VU
R T
DIAGRAM 7
F G
K
H
E J
DIAGRAM 8
T
R
QP
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Line And Planes In 3-Dimensions 14
9. Diagram 9 shows a right prism .
Name the angle between the plane ACS and theplane SADR.
A. CAS B. SCD
C. ACD D. CAD
10. Diagram 10 shows a right pyramid
MABCD with its square base ABCD .
The angle between the line MC and planeABCD is
A. MCD B. MCA
C. AMC D. AMD
11. The diagram 11 shows a right prism on a
horizontal plane. Given that STU is an
equilateral triangle and PV = VR = SW = WU .
The angle between the plane PRT and the plane
PRUS is
A. PQS B. RQU
C. TQW D. TVW
12. Diagram 12 shows a prism. A and B are
the mid-points of JK and ML, respectively.
JXK and MYL are isosceles triangle.
Name the angle between line AY and the planeJKLM.
A. YBA B. AYB
C. YAB D. ABY
DIAGRAM 11
DIAGRAM 9
Q C
DR
AS
P B
Q
P
y WS
U
R
T
y V
J K
L
Y
X
M
DIAGRAM 12
y A
y B
DIAGRAM 10
A B
CD
O
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Line And Planes In 3-Dimensions 15
13. Diagram 13 shows a right prism .
Name the angle between line DM and the planeACFD.
A. CDM B. DCM
C. CMD D. DCB
14. Diagram 14 shows a pyramid.
The angle between the plane GHJK and planeHJL is
A. GHL B. JHL
C. KJL D. KLJ
15. The diagram 15 shows a right prism on a
horizontal plane SRUT. Equilateral triangle
RUQ and STP are the uniform cross-section ofthe prism. M and N are the mid-points of STand RU, respectively.
The angle between the plane PRN and the
plane RSTU is
A. PRS B. MRP
C. PNM D. NPM
16. Diagram 16 shows a cuboid with base
PQRS . .
Name the angle between the plane WXR and
the plane QRWV.
A. QRX B. RXQ
C. RWX D. SXU
DIAGRAM 14DIAGRAM 13
C
D
y MA
B
F
L
JK
HG
R
U
S
P
T
Q
DIAGRAM 15
yN
y MR
Q
U
W
T
SP
V
y X
DIAGRAM 16
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Line And Planes In 3-Dimensions 16
17. Diagram 17 shows a cuboid with
rectangular base EFGH .
Name the angle between the plane PSG and the
plane PSHE.
A. SGH B. PGE
C. GSH D. GPE
18. Diagram 18 shows a cuboid on a
horizontal plane EFGH. J is a mid-point ofGH.
The angle between the plane ABGJ and the
plane ABCD is
A. AJE B. AGE
C. GBC D. GAC
19. Diagram 19 shows a prism on a horizontalplane QRP and vertical rectangular QRST .
Name the angle between the plane PRS and the
plane QRST is
A. PSQ B. PST
C. PRT D. PRQ
20. Diagram 20 shows a prism withrectangular base JKLM . LM is normal to baseJKLM .
Name the angle between line NK and base
JKLM.
A. NKM B. NMK
C. NMJ D. NKJ
DIAGRAM 17
E
H
R
P
Q
F
G
S
DIAGRAM 20
DIAGRAM 18
A
BC
D
E
F G
HJ
R
T
Q
P
S
DIAGRAM 19
J K
LM
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Line And Planes In 3-Dimensions 17
PAST YEAR SPM QUESTIONS
1. SPM Nov 2003, Q13
Diagram 6 shows a right prism with an isosceles triangle PQR as its horizontal base. M andN are the mid-point of SU and RQ, respectively.
Name the angle between the plane PQR and the plane PUS.
A. UPT B. NPT
C. PNQ D. MPN
2. SPM July 2004, Q16
Diagram 9 shows a cuboid with PQRS as its horizontal base.
Name the angle between the plane TQR and the plane TUVW.
A. TRW B. TQV
C. WTR D. VTQ
3. SPM Nov 2004, Q14
T
R
US y M
yN
P
P
Q
RS
WT
U V
DIAGRAM 6
DIAGRAM 9
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Line And Planes In 3-Dimensions 18
Diagram 8 shows a cuboid with horizontal base PQRS.
Name the angle between the line QV and the plane QUR.
A. VUR B. VUQ
C. VQU D. VQR
4. SPM July 2005, Q14
Diagram 6 shows a right pyramid NPQRS with square base PQRS.
The angle between the line NQ and the base PQRS is
A. PQN B. NQS
C. QNS D. QNR
5. SPM Nov 2005, Q14
P
RS
WT
U
Q R
SP
DIAGRAM 8
DIAGRAM 6
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Line And Planes In 3-Dimensions 19
Diagram 7 shows a right pyramid with a quadrilateral base EFGH.
What is the angle between the line VF and the base EFGH ?
A. VFE B. FVH
C. VFH D. FVE
6. SPM July 2006, Q14
Diagram 7 shows a cuboid with a horizontal base TUVW.
The angle between the line PW and the base TUVW is
A. PWV B. PUW
C. PTW D. PWU
E F
GH
V
DIAGRAM 7
V W
TU
SP
Q R
DIAGRAM 7
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Line And Planes In 3-Dimensions 20
7. SPM Nov 2006, Q14
Diagram 7 shows a cuboid with a horizontal base TUVW.
Diagram 7
Name the angle between the plane PQWT and the plane SRWT.
A QTR B QWR
C QTS
D QWS
8. SPM Jun 2007, Q 16
Diagram 9 shows a pyramid PQRS. The horizontal base QSR is a right angled triangle. Vertex P
is vertically above S.
Name the angle between the line PR and the plane PSQ.
A RPS
B RPQ
C PRS
D PRQ
P
Q
RS
Diagram
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Line And Planes In 3-Dimensions 21
9. SPM Nov 2007 Q14
Diagram 8 shows a pyramid with its rectangle base QRST.
Vertex P is vertically above T.
Name the angle between the plane PTS and the plane PTQ.
A PQT
B PST
C SPQ
D STQ
10. SPM Jun 2008, Q 15.
Diagram 8 shows a cuboid with a horizontal base TUVW.
What is the angle between the plane SVW and the plane PQRS?
A QSW
B RSW
C QSV
D PSV
P
Q
R
S
T
U
V
W
Diagram 8
P
Q
R
ST
Diagram 8
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Line And Planes In 3-Dimensions 22
11. SPM Nov 2008, Q14
Diagram 7 shows a right-angled triangular prism with the horizontal base QSTV.
What is the angle between the plane STU and the base QSTV?
A TUV
B UTV
C USV
D SUV
P
Q S
T
U
V
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Line And Planes In 3-Dimensions 23
ANSWERS
Exercise 1a) NUM b) NJG
c) SXT d) EBL
e) KMF f) APB
Exercise 2
a) VGO b) LSKc) RYT d) EQP
e) ENF f) RVS
17.2.4
1a) JKP or MLSb) UVP or XWS
2a) BAF or CDEb) URX or TSY
3a) VBC b) KPS c) PKJd) NHM e) ECD
4a) GLO b) PEF c) ELK
d) QNM e) TVW
Exercise 1
a) PQU or SRT
b) QJK or RML
c) XRU or YSTd) FBA e) MGH or LFE
f) ACB
Exercise 2
a) VLO b) MSJ or KRLc) YST d) EQP e) KEFf) UVP or XWS
Practice SPM Format
1. A 2. D 3.C 4. D 5. D
6. C 7. C 8.A 9. D 10. B11. D 12. C 13.A 14. C 15. C
16. A 17. C 18.C 19. D 20. A
SPM PAST YEAR QUESTIONS
1. D 2. C 3. C 4. B
5. C 6. A 7. B 8. A
9. D 10. B 11. B
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Line And Planes In 3-Dimensions 2
Problems Solving
The Angle Between The Line And The PlaneExercise 1 : Based on the diagram, calculate the angle between the line and the plane given
a) The diagram shows a cuboid. Calculate theangle between line NE and the plane of GFKN
b) The diagram shows a cuboid with ahorizontal base JKLM .Calculate the angle
between line KS and the plane of SRLM.
c) The diagram shows a prism. Calculate theangle between line RY and the plane of STY.
d) The diagram shows a prism. Calculate theangle between line QE and the plane of DCE.
J K
M
RS
L
6 cm
5 cm
cm
RS
U T
YX
cm
8 cm
14 cm
BA
F
E
D C
y Q
y P
6 cm
5 cm
12 cm
G H
EF
LK
N M12 cm
5 cm
16 cm
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Line And Planes In 3-Dimensions 3
e) The diagram shows a pyramid . Given that
HP = 13 cm. Calculate the angle between linePG and the plane of EHP.
f) The diagram shows a prism. Calculate the
angle between line UV and the plane of PSWV.
g) The diagram shows a pyramid with ahorizontal base DEFG. Given that VO = 9 cm.
Calculate the angle between line GV and theplane of DEFG.
h) The diagram shows a pyramid with atriangle base CHD. Calculate the angle
between line CA and the plane of ADH.
E F
GH
P
7 cm
9 cmP Q
RS
X
W
V
5 cm
4 cm
3 cm
7 cm
DE
FG
V
O
12 cm
5 cm
D
B
C
H
A
6 cm
8 cm
2 cm
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Line And Planes In 3-Dimensions 4
9. 2 Angle Between Two Planes
9.2.1 a) Calculate the angle between the two planes.
Example 1: Plane EFGH and
plane GHDA
Angle :
DHE = AGF
tan DHE =GF
AF
=6
9
DHE = 56.31o / 56
o19
1. a) Plane KLSP and planeJKLM
b) Plane PSWV and planeVUXW
Example 2 : Plane PQLK andplane SRLK
Angle :
QLR = PKS
tan QLR =LR
QR
=7
10
QLR = 55o
2. a) Plane ABCD and planeADEF
b) Plane URST and planeXRSY
G H
EF
DA
B C
8 cm
6 cm
P
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
P Q
RS
LK
12 cm
10 cm
7 cm
BA
F
E
D C
20 cm
10 cm
13 cm
RS
U T
YX
12
9 cm
5 cm
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Line And Planes In 3-Dimensions 5
Example 3 : Plane TRQ and
plane SRQP
Angle : TRS
tan TRS =RS
TS
=11
4
QLR = 19.98o / 19o59
3. a) Plane ABCD and plane
ABV
b) Plane PQSR and plane
PQKL
Example 4: Plane DEV andDEFG . VO = 7 cm
Angle : VMO
tan VMO =MO
VO
=6
7
VMO = 49.40o / 49o24
4a) Plane GCB and planeABCD
b) Plane PMNT and PlaneKLMN
A B
C
D
V
P Q
RS
LK
T
R
QP
AB
CD
G
O y L
5 cm
5 cm
11 cm
4 cm
8 cm
5 cm4 cm
3 cm
DE
FG
V
Oy M
10 cm
12 cm8 cm
12 cm
10 cm
T
L M
K
yF
P
9 cm
12 cm
10 cm
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Line And Planes In 3-Dimensions 6
Example 5 : Plane ABE and
plane ABCD
Angle : ELK
EK =22
915 = 12
tan ELK =LK
EK
=36
12
ELK =
4 a) Plane SRQ and plane
SRUT
b) Plane SURP and plane PTR
Exercise 1
a) The diagram shows a pyramid with ahorizontal 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 ahorizontal base JKLM .Calculate the angle
between the plane SRKJ and the plane of
SRLM.
BA
F
E
D
C
y L
y K
18 cm
15 cm
36 cm
R
S T
Q
yN
y M
8 cm
5 cm
P
y WU
R
T
y V
10 cm
4 cm
12 cm
J K
M
RS
L
BC
DA
V
O
10 cm
8 cm
7 cm
6 cm9 cm
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Line And Planes In 3-Dimensions 7
c) The diagram shows a prism. Calculate the
angle between the plane PLM and the plane ofPLNQ.
d) The diagram shows a prism. Calculate the
angle between the plane QRC and the plane ofPQRS.
e) The diagram shows a pyramid. Calculate
the angle between the plane FGP and the planeof EFGH
f) The diagram shows a prism. Name the angle
between the plane ABCD and the plane ofDQR.
KM
L N
QP
20 cm
10 cm
5 cm
P
D
C
S R
y A
y B
8 cm
10 cm
15 cm
E F
GH
P
18 cm
24 cm
14cm
P Q 14cm
RS
CD
A B
13 cm
7cm
9cm
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Line And Planes In 3-Dimensions 8
How to answer the SPM format Question
Example 1
Diagram 1 shows a pyramid LPQRS .
The base PQRS is a horizontal rectangle. J isthe midpoint of RS. The vertex L is 8 cm
vertically above the point J. Calculate the anglebetween 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 =
QR
SP
L
10 cm
12 cm
Diagram 1
J y
Q R
SP
L
10 cm
12 cm
J y
Q R
SP
L
10 cm
12 cm
J y
Q R
SP
L
10 cm
12 cm
J y
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Line And Planes In 3-Dimensions 9
Example 2
Diagram 2 shows a prism with horizontal
square ABCD. Trapezium KABL is the
uniform cross-section of the prism. Therectangular 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 planeNBC 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 / 36
o52
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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Line And Planes In 3-Dimensions 10
Questions Based on the Examination Format
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 andthe plane FEHG.
4. Diagram 4 shows a right prism with ahorizontal plane ABCD. It is a uniform prism
and its cross section is an isosceles triangle ofsides 4 cm. The thickness of the prism, EA = 4
cm.
Calculate the angle between the plane ABH
and the plane ABE.
DIAGRAM 1
DIAGRAM 2
DIAGRAM 3
A B
C
H
E D
DIAGRAM 4
K L
MN
RS
P Q
12 cm
4 cm
5 cm
P Q
RS
V
8 cm
6 cm
11 cm
F E
H
CD
A
G
7 cm
5 cm
6 cm
4 cm
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Line And Planes In 3-Dimensions 11
5) Diagram 5 shows a pyramid with the
horizontal plane, TRS. The rectangle PQRS isvertical plane.
Calculate the angle between the plane PTS and
the plane TQR.
6) Diagram 6 shows a cuboid. Z is the
midpoint of TW .
Calculate the angle between plane YVZ and thehorizontal plane XYVW.
7) Diagram 7 shows a right prism with basethe rectangular plane ABCD. Right triangle
BCF is the uniform cross-section of the prism.
The rectangular surface DCFE is vertical whilethe rectangular surface BAEF is inclined.
Calculate the angle between the plane DB andplane EDCF.
8) Diagram 8 shows a pyramid REFGH. Thebase 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 theplane EFGH.
Y V
WX
TS
RU
10 cm
6 cm
4 cm
y Z
DIAGRAM 6
T
R
QP
12 cm13 cm
10 cm
DIAGRAM 5
B
DIAGRAM 7 DIAGRAM 8
EF
GH
R
5 cm
24 cm
y S
A
CD
FE
8 cm
6 cm
6 cm
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Line And Planes In 3-Dimensions 12
9) Diagram 9 shows a cuboid. P is the midpoint
of line RQ.
Calculate the angle between the plane LQY andthe plane MQRN.
10) Diagram 10 shows a right prism. Right
angled triangle SUT is the uniform cross-section of the prism.
Calvulate the angle between the plane PSR andthe plane PUTR..
11) Diagram 11 shows a prism . The base
PQRS is a horizontal rectangle . X is themidpoint 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 arerectangle.
Calculate the angle between line LQ and the
base EFGH.
DIAGRAM 9
L M
QP
RS
K N y Y
10 cm
6 cm
12 cm
U
Q
ST
P
R
5 cm12 cm
20 cm
DIAGRAM 10
P Q
RS
ML
y X
12 cm
8 cm
5 cm
DIAGRAM 11
F
G
E
P
H
y M
y L
6 cm
5 cm
12 cm
DIAGRAM 12
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Line And Planes In 3-Dimensions 13
Past Year SPM Questions
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.
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]
K
F
D
HJE
L
6 cm
8 cm
Dia ram 1
A B
G
D C
E
F
H
12 cm
5 cm
cm
DIAGRAM 2
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Line And Planes In 3-Dimensions 14
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 isinclined.
Calculate the angle between the plane ABE and the base ABCD. [3 marks]
B
E
A
CD
F
12
5 cm
3
DIA RAM
DIAGRAM 2
K J
ML
V
10 cm
12
Q y
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Line And Planes In 3-Dimensions 15
5. Nov 2005, Q4
Diagram 1 shows a right prism. Right angled triangled PQR is the uniform cross-section ofthe 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 angledtriangle NHJ is the uniform cross-section of the prism.
Identify and calculate the angle between the line KN and the plane HLMN.
U
Q
ST
P
R
12
5
1
DIAGRAM 1
DIAGRAM 2
J
M
H
KL
N
6 cm
12
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Line And Planes In 3-Dimensions 16
7. Nov 2006, Q2
Diagram 1 shows a right prism. The base PQRS is on horizontal rectangle. The right triangleUPQ 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 planeand the rectangle UVWT is an inclined plane.
Identify and calculate the angle between the plane PQW and the base PQRS.[3 marks]
P R
W
Q
12 cm
T
S
U
7 cm
14 cm
5 cm
V
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Line And Planes In 3-Dimensions 17
9. SPM Nov 2007 Q4Diagram 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 2008Diagram shows a cuboid ABCDEFGH with horizontal base ABCD. P, Q and R are the
midpoints of BC, AD and FE respectively.
Name and calculate the angle between the plane FPCR and the base ABCD.
[4 marks]
P
Q
R
S
T
U
V
16 cm
12 cm
5 cm
A
B
C
D
E
F
G
H
P
R
68
5
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Line And Planes In 3-Dimensions 18
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]
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 /38
o40
3e 18.43o / 18o26
Exercise 1a 50.91
o /
50o54
b 26.57o / 26
o34 c 54.46o / 54
o28 d 71.57o /
71o34
e 28.30o /28o18
f 51.34o / 51o20 g 54.16o / 54o10 h 53.13o /53o8
9.2.11a 36.87
o /36
o52
1b 74.05o / 74o3 2a 67.38
o / 67o23 2b 29.05
o /29
o3
3a 57.99o / 58o 3b 36.89o / 36o52 4a 60o 4b 53.13o /53o8
5a 36.87o
/36
o52
5b 63.43o
/ 63o26
Exercise 1
a 66.04o /66
o2 b 33.69o / 33o41 c 26.57o /
26o34d 66.42o /
66o25
e 37.87o
/37o52
f 34.70o
/ 34o42
PRACTICE SPM FORMAT
1 47.73o
/47o44
2 17.10o
/ 17o6 3 54.46
o/
54o284 56.31
o/
56o19
5 63.43o /
63o26
6 36.87o / 36
o52 7 36.87
o /
36o52
8 34.70o /
34o429 30.96
o /
30o58
10 30.96o / 30o58 11 53.13
o /
53o8
12 18.43o /
18o26SPM PAST YEAR QUESTIONS
1 Nov 2003 36.87o / 36o52
2 Jul 2004 18.43o / 18
o 26
3 Nov 2004 31.61o / 31o 36
4 Jul 2005 14.04o
/ 14o
2
5 Nov 2005 33.69o / 33o 41
6 Jul 2006 50.19o
/ 50o
127 Nov 2006 34.70 / 34
O42
8 Jun 2007 ,54.46 or54 28'WQR r r
9 Nov 2007 SUV , 31.61 or 31 36'r r
10 Jun 2008 ,32QPR r
11 Nov 2008 ,15.47 or 15 28'EAM r r