Chapter 17 I Lines & Planes in 3D ENHANCE
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Transcript of Chapter 17 I Lines & Planes in 3D ENHANCE
8/3/2019 Chapter 17 I Lines & Planes in 3D ENHANCE
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Line And Planes In 3-Dimensions 1
CHAPTER 17 : LINES AND PLANES IN 3 DIMENSIONS
Definitions
11.1 The angle between a line and a plane is the angle between the line and its orthogonal
projection on the plane.
11.2 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 D
raw 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.
P Q
R S
K
L
V
o
Line: KV
Plane : PQR S
Normal to the plane: VLOrthogonal projection of the line
onto the plane: KL
Angle between KV and the plane
PQR S is VK L
LM andMN are perpendicular toDC.
The angle between the plane
ABCD and the plane CDEF is
K ML ( or ADE or BCF )
M
L
N A B
C D
E F
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Line And Planes In 3-Dimensions 2
17.1 Angle Between Lines And Planes
17.1.1 Draw and /or colour the given line
Example 1 : Line AH 1. a) Line BE b) Line FD
Example 2: Line PL 2.a) Line QS b) Line RK
Example 3: Line VF 3. a) Line VM b) Line V N
Example 4 : Line DV 4. a) Line AC b) Line BV
G H
E F
D A
B C
G H
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
P Q
R S
LK
P Q
R S
LK
D E
F G
V
O
D E
F G
V
O
D E
F G
V
OMy
Ny
A B
C D
V
A B
C D
V
A B
C D
V
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Line And Planes In 3-Dimensions 3
17.1.2 Shade Or Colour The Plane Given
Example 1: Plane EFGH 1. a) Plane DEFG b) Plane PQR S
Example 2 : Plane PSK 2. a) Plane ABGF b) Line BCV
Example 3 : Plane DEV 3. a) Plane ADHG b) Plane ADV
c) Plane ACV d) Plane VGO e) Line K SQ
G H
E F
D A
B C
G H
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
P Q
R S
LK
P Q
R S
LK
D E
F G
V
O
D E
F G
V
O
D E
F G
V
O
A B
C D
V
A B
C D
V
A B
C D
V
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Line And Planes In 3-Dimensions 4
17.1.3 Complete the following question
i) shade the plane givenii) draw the line given
iii) determine the point which lies on the plane given
Example 1: Plane :EFGH
Line :GD
1. a) Plane : DEFGLine : DV
b) Plane : PQR S Line : KQ
Example 2 : Plane : PSK Line : KR
2. a) Plane : CDEHLine : GC
b) Plane : BCVLine : AV
Example 3 : Plane : GEV
Line : VF
3. a) Plane : CDV
Line : BV
b) Plane : ADEF
Line : DG
c) Plane : ABCD
Line : AV
d) Plane : DVF
Line : EV
e) Plane : K SP
Line : KQ
G H
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
P Q
R S
LK
E D
F G
V
O
D E
F G
V
OA B
C D
V
A B
C D
V
A B
C D
V
G H
E F
D A
B C
D E
F G
V
O
P Q
R S
LK
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Line And Planes In 3-Dimensions 5
17.1.4 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 : PQR S Line : KQ
Normal line :
Orthogonal projection :
Example 2 : Plane : PSK
Line : KR
Normal line : SR Orthogonal 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 : ADEF Line : DG
Normal line :Orthogonal projection :
G H
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
E D
F G
V
O A B
C D
V
A B
C D
V
G H
E F
D A
B C
D E
F G
V
O
P Q
R S
LK
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Line And Planes In 3-Dimensions 6
17.1.5 (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 : PQR S
Line : KQ
Angle :
Example 2 : Plane : PSK Line : KR
Angle : RK S
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
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
E D
F G
V
OA B
C D
V
A B
C D
V
G H
E F
D A
B C
D E
F G
V
O
P Q
R S
LK
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Line And Planes In 3-Dimensions 7
Exercise 1 : Name the angle between the line and the plane given
a) Line U N and plane PQU b) Line J N 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 PQR S
J K
Q
M
R S
P
L
y N
yG
T
P Q
R S y N
yM
U
R S
U T
YX
BA
F
E
D C
G H
E F
LK
N M
P Q
R S
X
W
V
Ay
By
yL
yK
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Line And Planes In 3-Dimensions 8
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 K S 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 GFK N
f) The diagram shows a prism. Name the angle between line RV and the plane of PSWV.
J K
M
R S
P
L
R
S
U T
YX
BA
F
E
D C
G H
E F
LK
N M
P Q
R S
X
W
V
U
yQ
y P
D E
F G
V
O
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Line And Planes In 3-Dimensions 9
17. 2 Angle Between Two Planes
17.2.1 Shade or colour the planes given
Example 1 :
Plane GHEF and plane BHEA
1. a) Plane ABCD and plane
ABHE
b) Plane AGHD and plane
ABCD
Example 2:Plane PLR and QRL
2.a) Plane PSK and planeK SQ
b) PlaneSRLK and planeKLP
Example 3:
Plane VEF and plane DEFG
3. a) Plane DVG and plane
VEF
b) Plane VDE and plane
DEFG
Example 4 :
Plane ABV and plane ABCD
4. a) Plane AVC and plane
VBC
b) Plane VDC and plane VBC
G H
E F
D A
B C
G H
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
P Q
R S
LK
P Q
R S
LK
D E
F G
V
OD
E
F G
V
O
D E
F G
V
O
Ny
A B
C D
V
A B
C D
V
A B
C D
V
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Line And Planes In 3-Dimensions 10
17.2.1 Name the line of intersection between the two planes
Example 1 :
Plane GHEF and plane BHEA
Line : HE
1. a) Plane ABCD and plane
ABHE
Line : «««..
b) Plane AGHD and plane
ABCD
Line : «««
Example 2:
Plane PLR and QRL
Line : RL
2.a) Plane PSK and plane
K SQ
Line : ««««
b) PlaneSRLK and plane
KLP
Line : ««««
Example 3:
Plane VEF and plane DEFG
Line : EF
3. a) Plane DVG and plane
DEFG
Line : ««««
b) Line VDE and plane
DEFG
Line : ««««
Example 4 :
Plane ABV and plane ABCD
Line : AB
4. a) Plane DAV and plane
DAC
Line : ««««
b) Plane VDC and plane VBC
Line : ««««
G H
E F
D A
B C
G H
E F
D A
B C
G H
E F
D A
B C
P Q
R S
LK
P Q
R S
LK
D E
F G
V
O
D E
F G
V
O
Ny
A B
C D
V
A B
C D
V
P Q
R S
LK
D E
F G
V
O
A B
C D
V
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Line And Planes In 3-Dimensions 11
17.2.2 Name and colour the perpendicular line to the line intersection between the two planes
Example 1: Plane EFGH and plane GHDA
Line EH and DH or line
FG and AG are
perpendicular to GH
1. a) Plane KLSP and plane
JKLM
b) Plane PSWV and plane
VUXW
Example 2 : Plane PQLK and
plane SRLK
Line PK and SK or line
QL and RL are
perpendicular to K L
2. a) Plane ABCD and plane
ADEF
b) Plane UR ST and plane
XR SY
Example 3 : Plane TRQ and plane SRQP
Line TR and SR is
perpendicular to QR
3. a) Plane ABCD and planeABV
b) Plane PQSR and planePQK
A B
C D
V
J K
Q
M
R S
P
L
P Q
R
X
W
U
G H
E F
D A
B C
BA
F
E
D C
R S
U T
YX
P Q
R S
LK
P Q
R S
LK
T
R
QP
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Line And Planes In 3-Dimensions 12
c) Plane JKLM and plane
PKL
d) Plane MHEL and plane
NHE
e) Plane ABCD and plane
BCE
Example 4: Plane DEV and
DEFG
Line VM and OM is
perpendicular to DE
4a) Plane GCB and plane
ABCD
b) Plane KLMN and plane
KP N
c) Plane ABE and plane
ABCD
d) Plane RUQ and plane
SRUT
e) Plane SURP and plane PTR
J K
Q
M
R S
P
L
BA
F
E
D C
G H
E F
LK
N M
D E
F G
V
O
yM A B
C D
G
O yLT
LM
N K
yE
yF
P
BA
F
E
D C
yL
yK
R
S
T
Q
y N
yM
P
S
Q
yWU
R
T
yV
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Line And Planes In 3-Dimensions 13
17.2.3 a) Name the angle between the two planes.
Example 1: Plane EFGH and plane 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 PK S
2. a) Plane ABCD and plane
ADEF
b) Plane UR ST and plane
XR SY
Example 3 : Plane TRQ and
plane SRQP
Angle : TRS
3. a) Plane ABCD and plane
ABV
b) Plane PQSR and plane
PQK
A B
C D
V
J K
Q
M
R S
P
L
P Q
R
X
W
U
G H
E F
D A
B C
BA
F
E
D C
R S
U T
YX
P Q
R S
LK
P Q
R S
LK
T
R
QP
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Line And Planes In 3-Dimensions 14
c) Plane JKLM and planePKL
d) Plane MHEL and plane NHE
e) Plane ABCD and planeBCE
Example 4: Plane DEV andDEFG
Angle : VMO
4a) Plane GCB and planeABCD
b) Plane KLMN and planeKP N
c) Plane ABE and plane
ABCD
d) Plane RUQ and plane
SRUT
e) Plane SURP and plane PTR
J K
Q
M
R S
P
L
BA
F
E
D C
G H
E F
LK
N M
D E
F G
V
O
yM AB
C D
G
O yL
BA
F
E
D C
yL
yK
R
S
T
Q
y N
yM
T
LM
N K
yE
y F
P
Q
yW
R
T
yV
S
P
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Line And Planes In 3-Dimensions 15
Exercise 1 : Name the angle between the two planes
a) Plane SRQP and plane QUTR b) Plane JQR M and plane JKLM
c) Plane R SYX and plane UR ST d) Plane BCF and plane ABCD
e) Plane GMLF and plane GHEF f) Plane PQA and plane PQR S
C y
J K
Q
M
R S
P
L
R S
U T
YX
BA
F
E
D C
G H
E F
LK
N M
P Q
R S
X
W
V
Ay
By
T
Q
R
U
P
S
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Line And Planes In 3-Dimensions 16
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 of
DEFG.
b) The diagram shows a cuboid with a
horizontal base JKLM . Name the angle
between the plane SRKJ and the plane of
SRLM.
c) The diagram shows a prism. Name theangle between the plane R SY and the plane of
R STU.
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 angle between the plane UVWX and the plane of
PSWV.
J K
M
R S
P
L
R S
U T
YX
BA
F
E
D C
G H
E F
LK
N M
yQ
y P
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Line And Planes In 3-Dimensions 17
Questions According To SPM Format
1. Diagram 1 shows a pyramid with a
rectangular base PQR S. V is vertically above P.
Name the angle between the line VR and the
plane PQR S
A. PRV B. VR S
C. PVQ D. VQS
2. Diagram 2 shows a cuboid.
Name the angle between the line PM and the
plane SR MN.
A. PMK B. PMQ
C. PMR D. PMS
3. In the diagram 3, X and Y are the midpointsof PW and SV respectively.
The angle between line RX and the plane
R SVU 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. EK F
C. HEK D. JEK
K L
M N
R S
P Q
P Q
R S
V
DIAGRAM 1 DIAGRAM 2
Q P
WS
VU
R T
yX
DIAGRAM 3
yY
F G
K
H
E J
DIAGRAM 4
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Line And Planes In 3-Dimensions 18
5. Diagram 5 shows a pyramid .PQR S is arectangle.
Name the angle between the line TQ and the
plane PQR S
A. QPT B. QST
C. R ST D. SQT
6. Diagram 6 shows a cuboid.
The angle between the line SU and plane
PSWT 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 and
the plane FGKJ.
A. EGF B. EK F
C. HGJ D. GK E
T U
VW
R S
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 19
9. Diagram 9 shows a right prism .
Name the angle between the plane ACS and the plane SADR.
A. CAS B. SCD
C. ACD D. CAD
10. Diagram 10 shows a right pyramidMABCD 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 anequilateral triangle and PV = VR = SW = WU .
The angle between the plane PRT and the planePRUS 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 plane
JKLM.
A. YBA B. AYB
C. YAB D. ABY
DIAGRAM 11
DIAGRAM 9
Q C
D R
AS
P B
Q
P
yWS
R
T
yV
J K
L
Y
X
M
DIAGRAM 12
yA
yB
DIAGRAM 10
A B
C
O
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Line And Planes In 3-Dimensions 20
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 triangleRUQ and STP are the uniform cross-section of the prism. M and N are the mid-points of ST
and RU, respectively.
The angle between the plane PR N and the plane R STU is
A. PR S B. MRP
C. P NM D. NPM
16. Diagram 16 shows a cuboid with base
PQR S . .
Name the angle between the plane WXR andthe plane QRWV.
A. QRX B. RXQ
C. RWX D. SXU
DIAGRAM 14DIAGRAM 13
C
D
yM A
B
F
L
J
HG
R
U
S
P
T
Q
DIAGRAM 15
y N
yM R
Q
U
W
T
S
P
V
yX
DIAGRAM 16
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Line And Planes In 3-Dimensions 21
17. Diagram 17 shows a cuboid withrectangular 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 ahorizontal plane EFGH. J is a mid-point of
GH.
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 horizontal plane QRP and vertical rectangular QR ST .
Name the angle between the plane PR S and the plane QR ST is
A. PSQ B. PST
C. PRT D. PRQ
20. Diagram 20 shows a prism withrectangular base JKLM . LM is normal to base
JKLM .
Name the angle between line NK and base
JKLM.
A. NK M 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
L
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Line And Planes In 3-Dimensions 22
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 and
N 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. P NQ D. MP N
2. SPM July 2004, Q16
Diagram 9 shows a cuboid with PQR S as its horizontal base.
Name the angle between the plane TQR and the plane TUVW.
A. TRW B. TQV
C. WTR D. VTQ
T
R
US yM
y N
P
P Q
R S
WT
U V
DIAGRAM 6
DIAGRAM 9
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Line And Planes In 3-Dimensions 23
3. SPM Nov 2004, Q14
Diagram 8 shows a cuboid with horizontal base PQR S.
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 NPQR S with square base PQR S.
The angle between the line NQ and the base PQR S is
A. PQ N B. NQS
C. Q NS D. Q NR
P
R S
WT
U
Q R
S P
N
DIAGRAM 8
DIAGRAM 6
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Line And Planes In 3-Dimensions 24
5. SPM Nov 2005, Q14
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
S P
Q R
DIAGRAM 7
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Line And Planes In 3-Dimensions 25
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 PQR S. 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 R PS
B R P Q
C P RS
D P RQ
P
Q
R S
Diagram
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Line And Planes In 3-Dimensions 26
9. SPM Nov 2007 Q14
Diagram 8 shows a pyramid with its rectangle base QR ST.
Vertex P is vertically above T.
Name the angle between the plane PTS and the plane PTQ.
A P QT
B PS T
C SP Q
D S TQ
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 PQR S?
A QSW
B RSW
C QS V
D PS V
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 27
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 U S V
D S UV
P
Q S
T
U
V
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Line And Planes In 3-Dimensions 28
A NSWER S
17.1.21a) VO , DO b) K S , SQ
2a) GH , CH b) AB , BV
3a) BC , VC b) GF , FD
17.1.3
1a) VDO b) KQS 2a) GCH b) AVB
3a) BVC b) FDG
Exercise 1a) NUM b) NJG
c) SXT d) EBL
e) K MF f) APB
Exercise 2
a) VGO b) LSK c) RYT d) EQP
e) ENF f) RVS
17.2.21a) AB b) AD
2a) K S b) KL
3a) DG b) DE 4a) AD b) VC
17.2.31a) Line SL and ML or line PK and JK are
perpendicular to LK
b) Line UV and PV or line XW and SW are perpendicular to VW
2a) Line FB and FA or line CD and ED are perpendicular to AD
b) Line XR and UR or line ST and YS are perpendicular to R S
3a) Line VB and CB is perpendicular to AB
b) Line KP andSP is perpendicular to PQ
c) Line PK and JK is perpendicular to KL
d) Line NH and MH is perpendicular to EH
e) Line EC andDC is perpendicular to BC
4a) Line GL and OL is perpendicular to BC
b) Line PE and EF is perpendicular to LM
c) Line EL and LK is perpendicular to AB
d) Line Q N andMN is perpendicular to RU
e) Line WV and TV is perpendicular to PR
17.2.4
1a) JKP or MLS
b) UVP or XWS
2a) BAF or CDE
b) URX or TSY
3a) VBC b) KPS c) PKJ
d) NHM e) ECD
4a) GLO b) PEF c) ELK
d) Q NM e) TVW
Exercise 1
a) PQU or SRT b) QJK or R MLc) XRU or YST
d) FBA e) MGH or LFE f) ACB
Exercise 2
a) VLO b) MSJ or KRL
c) YST d) EQP e) K EF f) 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. B
11. 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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