Mathematics Form 4 ACTIVITY SHEET Pupil’s Copy Lesson 104: Horizontal Planes, Vertical Planes and Inclined Planes.
J
Question 1 I
E F D A
H C
G
B
Based on the figure given above, identify the type of planes for: Plane
Type of planes
EDIJ ABGF BGHC ABCDE CHID
F
Question 2
C D
E B A
Based on the figure given above, identify the type of planes for: Plane
Type of planes
ABCD AEB BCFE
1
Mathematics Form 4
F
Question 3
D
C E
A
B
Based on the figure given above, identify the type of planes for: Plane
Type of planes
AFB CFD ABCD
Question 4 H
G F
E
C
D A
B
Based on the figure given above, identify the type of planes for: Plane
Type of planes
ABFE EFGH ABCD
2
Mathematics Form 4
ACTIVITY SHEET Teacher’s Copy Lesson 104: Horizontal Planes, Vertical Planes and Inclined Planes.
J
Question 1 I
E F D A
H C
G
B
Based on the figure given above, identify the type of planes for: Plane
Type of planes
EDIJ
Inclined plane
ABGF
Horizontal plane
BGHC
Vertical plane
ABCDE
Vertical plane
CHID
Horizontal plane
F
Question 2
C D
E B A
Based on the figure given above, identify the type of planes for: Plane
Type of planes
ABCD
Vertical plane
AEB
Horizontal plane
3
Mathematics Form 4 BCFE
Vertical plane
F
Question 3
D
C E
A
B
Based on the figure given above, identify the type of planes for: Plane
Type of planes
AFB
Inclined plane
CFD
Inclined plane
ABCD
Horizontal plane
Question 4 H
G F
E
C
D A
B
Based on the figure given above, identify the type of planes for: Plane
Type of planes
ABFE
Vertical plane
EFGH
Horizontal plane
ABCD
Horizontal plane
4
Mathematics Form 4 ACTIVITY SHEET Pupils’ Copy Lesson 105: Lines on a Plane, Lines Intersect with a Plane and Normals. 1. Name all the lines that lie on the plane CDHG.
Question 1 F
G H
E B
C
A
D
Answer: 2. Name all the lines that intersect with the plane BCGF. Answer : 3. Name all the normals to the plane ABFE Answer :
Question 2
1. Name all the lines that lie on the plane RQTU.
S
U
Answer:
W R
P
2. Name all the lines that intersect with the plane PQTS. Answer : T
3. Name all the normals to the plane STU
V
Answer :
Q
1. Name all the lines that lie on the plane ABHG.
Question 3 K E
J
Answer:
D L
F
C
I
Answer :
G H
A
3. Name all the normals to the plane ABCDEF
B
Answer : 1. Name all the lines that lie on the plane ABCD.
Question 4 C
G
Answer: D
H O
K
N M
2. Name all the lines that intersect with the plane LKOP.
L
P
E
2. Name all the lines that intersect with the plane FAGL.
Answer :
J I
F A
B
3. Name all the normals to the plane EFGH Answer :
1
Mathematics Form 4 ACTIVITY SHEET Teacher’s Copy Lesson 105: Lines on a Plane, Lines Intersect with a Plane and Normals. 1. Name all the lines that lie on the plane CDHG.
Question 1 F
G H
E B
C
A
D
Answer: Line: CD, DH, HG, GC, DG 2. Name all the lines that intersect with the plane BCGF. Answer : Line: BA, CD, GH, FE, GD 3. Name all the normals to the plane ABFE Answer : Line: AD, BC, FG, EH
Question 2
1. Name all the lines that lie on the plane RQTU.
S
U
Answer: Line: RQ, QT, TU, UR, WV
W R
P
2. Name all the lines that intersect with the plane PQTS. Answer : Line: QR, VW, TU, PR, SU T
3. Name all the normals to the plane STU
V
Answer : Line: SP, UR, TQ
Q
1. Name all the lines that lie on the plane ABHG.
Question 3 K E
J
Answer: Line: AB, BH, HG, GA
D L
F
C
I
Answer : Line: AB, GH, LK, FE
G H
A
3. Name all the normals to the plane ABCDEF
B
Answer : Line: AG, BH, CI, DJ, EK, FL 1. Name all the lines that lie on the plane ABCD.
Question 4 C
G
Answer: Line: AB, BC, CD, DA, IJ, JK, KL, LI D
H O
N M
2. Name all the lines that intersect with the plane LKOP.
K L
P
E
2. Name all the lines that intersect with the plane FAGL.
Answer : Line: LI, KJ, PM, ON
J I
F A
B
3. Name all the normals to the plane EFGH Answer : Line: EA, FB, GC, HD
2
Mathematics Form 4 ACTIVITY SHEET Pupil’s Copy LESSON 106: Orthogonal projection of a line on a plane.
1. The figure shows a triangular prism PQRSTU. Draw and name the orthogonal projection of each of the following lines on the plane PQRS.
U
S
T
P
M
N
Q
a) TQ
b) QU
1
R
Mathematics Form 4
2. The figure shows a cube ABCDEFGH. Based on the diagram given, complete the table below.
H
G
E
F D
C
A
B
Line
Plane
1
AG
ABCD
2
HB
ABEF
3
CH
EFGH
4
EG
DCHG
5
HB
BCFG
Orthogonal Projection
2
Angle between line and plane
Mathematics Form 4
3. The figure shows a pyramid ABCDEFG. Based from diagram given, complete the table below.
E
D
G C A
No
Line
B
F
Plane
Angle between line and plane
1
ABCD
EFG or GFE
2
ABCD
EAC or CAE
3
ABCD
EBD or DBE
4
ABCD
EGF or FGE
5
ABCD
EDB or BDE
3
Mathematics Form 4
ACTIVITY SHEET Teacher’s Copy LESSON 106: Orthogonal projection of a line on a plane.
1. The figure shows a triangular prism PQRSTU. Draw and name the orthogonal projection of each of the following lines on the plane PQRS.
U
S
T
P
M
R
N
Q T
Q
M a) TQ
MQ or QM
U
Q
N b) QU
NQ or QN
4
Mathematics Form 4
2. The figure shows a cube ABCDEFGH. Based on the diagram given, complete the table below.
H
G
E
F D
C
A
B
Line
Plane
Orthogonal Projection
Angle between line and plane
1
AG
ABCD
AC
GAC
2
HB
ABEF
BF
HBF
3
CH
EFGH
HG
CHG
4
EG
DCHG
HG
EGH
5
HB
BCFG
FB
HBF
5
Mathematics Form 4
3. The figure shows a pyramid ABCDEFG. Based from diagram given, complete the table below.
E
D
G C A
B
F
No
Line
Plane
Angle between line and plane
1
EF
ABCD
EFG or GFE
2
EA
ABCD
EAC or CAE
3
EB
ABCD
EBD or DBE
4
EG
ABCD
EGF or FGE
5
ED
ABCD
EDB or BDE
6
Mathematics Form 4
ACTIVITY SHEET Pupil’s Copy LESSON 107: Problems Solving on Angle between a Line and a Plane. The diagram shows a cuboid. KG = 2HK. DL = HK. By sketching an appropriate right-angled triangle, find the following: (a) the length of FD.
K
H E
G F
L
D
4 cm
C 6 cm
A
(b) the length of KA .
(c) the angle between FD and BCGF.
(d) the angle between KA and EFGH.
1
9 cm
B
Mathematics Form 4 ACTIVITY SHEET Teacher’s Copy LESSON 107: Problems Solving on Angle between a Line and a Plane. The diagram shows a cuboid. KG = 2 HK. DL = HK. By sketching an appropriate rightangled triangle, find the following:
K H
(a) the length of FD.
FD =
E
BF 2 + BD 2
= 16 + 9 + 6
A
L
D
C B
9 cm
F
2
= 133 = 11.53 cm
4 cm
F
6 cm
= 4 2 + ( AB 2 + AD2 ) 2
G
B
D
(b) the length of KA .
KA = KL2 + AL2
K
= 42 + ( AD 2 + DL2 )
= 16 + 62 + 32
A
L
= 7.81 cm (c) the angle between FD and BCGF. The orthogonal projection of FD on BCGF is FC. So, the angle between FD and BCGF is ∠ CFD. D CD sin ∠ CFD = DF
9 cm
C
11.53 cm
9 = 11.53
F
= 0.7806 ∠ CFD = 51.32° or 51° 19' (d) the angle between KA and EFGH. The orthogona l projection of KA on EFGH is KE. So, the angle between KA and EFGH is ∠ EKA.
AE AK 4 = 7 .81 = 0.5122 ∠ EKA = 30.81° or 30° 49'
sin ∠EKA =
K
E 4 cm
A
2
7.81 cm
Mathematics Form 4 ACTIVITY SHEET Pupil’s Copy Lesson 108: Line of Intersection between Two Planes.
J
Question 1 I
E F D A
H C
G
B
Based on figure given above, state the line of intersection between the following planes: Plane A
Plane B
EDIJ
DCHI
BFIC
ABGF
AGH
BCHG
ABCDE
DEJI
DCG
BCHG
Line of intersection
F
Question 2
C D
E B A
Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection. Plane A
Plane B
ABCD
BCFE
AFE
BCFE
BED
ABE
Line of intersection
1
Pair of lines perpendicular to line of intersection
Mathematics Form 4
F
Question 3
D
C E
A
B
Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection. Plane A
Plane B
AFC
ABCD
BEF
BCE
BDF
ABCD
Pair of lines perpendicular to line of intersection
Line of intersection
Question 4 H
G F
E
C
D A
B
Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection Plane A
Plane B
AEGC
ABFE
BHD
ADHE
EFGH
ABFE
Line of intersection
2
Pair of lines perpendicular to line of intersection
Mathematics Form 4 ACTIVITY SHEET Teacher’s Copy Lesson 108: Line of Intersection between Two Planes.
J
Question 1 I
E F D A
H C
G
B
Based on figure given above, state the line of intersection between the following planes: Plane A
Plane B
Line of intersection
EDIJ
DCHI
Line DI
BFIC
ABGF
Line BF
AGH
BCHG
Line GH
ABCDE
DEJI
Line ED
DCG
BCHG
Line CG
F
Question 2
C D
E B A
Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection. Plane A
Plane B
Line of intersection
ABCD
BCFE
Line BC
Pair of lines perpendicular to line of intersection Line AB & Line BE or Line CD & Line CF
AFE
BCFE
Line EF
Line EA & Line EB
BED
ABE
Line BE
Line EA & Line ED
3
Mathematics Form 4
F
Question 3
D
C E
A
B
Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection. Plane A
Plane B
Line of intersection
AFC
ABCD
Line AC
Pair of lines perpendicular to line of intersection Line EF & Line BD
BEF
BCE
Line BE
Line EC & Line EF
BDF
ABCD
Line BD
Line EF & Line AC
Question 4 H
G F
E
C
D A
B
Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection Plane A
Plane B
Line of intersection
AEGC
ABFE
Line AE
Pair of lines perpendicular to line of intersection Line EF & Line EH or Line AB & Line AD
BHD
ADHE
Line DH
Line AD & Line BD
EFGH
ABFE
Line EF
Line EA & Eh or Line FB & Line FG
4
Mathematics Form 4 ACTIVITY SHEET Pupil’s Copy LESSON 109: Angle between Two Planes.
1. Based on the diagram given, identify the angle between the two planes given by completing the table below.
R
S
O
P
Q
T
U
M
N
No
Plane
Plane
1
UON
MNPQ
2
ROQ
MNPQ
3
NORQ
MORP
4
RSU
MORP
5
TQU
NORQ
1
Angle between two p lanes
Mathematics Form 4 2. The figure shows a cuboid ABCDPQRS. Complete the table given in order to identify the the angle between two planes.
D
C R
S
A
P
B Q
No
Plane
Plane
1
CDS
ABCD
2
PQB
PQRS
3
ABC
ABPQ
4
SRP
DCSR
5
ABPQ
CBRQ
2
Angle between two planes
Mathematics Form 4 3. The figure shows a pyramid ABCDEFG. Based from diagram given, complete the table below.
E
G
D
C
I
H A
No
Plane
B
F
Plane
Angle between two planes
1
ABCD
EFG or GFE
2
ABCD
EGF or FGE
3
ABCD
EHI or IHE
4
ABCD
EIH or HIE
3
Mathematics Form 4 ACTIVITY SHEET Teacher’s Copy LESSON 109: Angle between Two Planes.
1. Based on the diagram given, identify the angle between the two planes given by completing the table below.
R
S
O
P
Q
T
U
M
N
No
Plane
Plane
Angle between two planes
1
UON
MNPQ
ONM or MNO
2
ROQ
MNPQ
RQP or PQR
3
NORQ
MORP
UST or TSU, NOM or MON, QRP or PRQ
4
RSU
MORP
UST or TSU
5
TQU
NORQ
SUT or TUS
4
Mathematics Form 4 2. The figure shows a cuboid ABCDPQRS. Complete the table given in order to identify the the angle between two planes.
D
C R
S
A
P
B Q
No
Plane
Plane
Angle between two planes
1
CDS
ABCD
SDA or ADS
2
PQB
PQRS
RQB or BQR
3
ABC
ABPQ
CBQ or QBC
4
SRP
DCSR
DSP or PSD
5
ABPQ
CBRQ
RQP or PQR, CBA or ABC
5
Mathematics Form 4 3. The figure shows a pyramid ABCDEFG. Based from diagram given, complete the table below.
E
G
D
C
I
H A
B
F
No
plane
Plane
Angle between two planes
1
EAB
ABCD
EFG or GFE
2
ECD
ABCD
EGF or FGE
3
EAD
ABCD
EHI or IHE
4
ECB
ABCD
EIH or HIE
6
Mathematics Form 4 ACTIVITY SHEET Pupil’s Copy Lesson 110: Problems Solving on Lines and Planes in 3-Dimensional Shapes. E 20cm F
Question 1 55cm
B
A
L
40cm
M 15cm
H
50cm
D
I
G
J
N
K
C
The figure shows two wooden blocks attached together. Based on the figure, solve the questions given: 1. What is the length of the diagonal DI? 2. Determine the angle between planes ILNK and CILG. 3. Determine the angle between horizontal plane DCGH and the diagonal DF. 4. Determine the angle between planes LGKJ and CGFB. F
Question 2 B
N
D
A
8cm M 16cm
E
The figure shows a prism. M and N are the midpoints of AB and DE respectively.
16cm B
1. What is the length of the diagonal AE? 2. Identify the angle between planes ACFD and ABED. 3. Determine the angle between the diagonal BF and the plane ABED. 1
Mathematics Form 4 ACTIVITY SHEET Teacher’s Copy Lesson 110: Problems Solving on Lines and Planes in 3-Dimensional Shapes. E 20cm F
Question 1 55cm
B
A
L
40cm
M 15cm
H
50cm
D
I
G
J
N
K
C
The figure shows two wooden blocks attached together. Based on the figure, solve the questions given: 1. What is the length of the diagonal DI?
25cm
2. Determine the angle between planes ILNK and CILG.
69.44°
3. Determine the angle between horizontal plane DCGH and the diagonal DF.
42.0°
4. Determine the angle between planes LGKJ and CGFB.
36.02°
F
Question 2 B
N
D
A
8cm M 16cm
E
The figure shows a prism. M and N are the midpoints of AB and DE respectively.
16cm B
1. What is the length of the diagonal AE?
18.87cm
2. Identify the angle between planes ACFD and ABED.
58.0°
3. Determine the angle between the diagonal BF and the plane ABED.
25.52°
2
