This document provides a summary of a 2 hour mathematics enrichment session on lines and planes in 3-dimensions. It includes 10 problems involving calculating angles between lines and planes using trigonometric ratios. The problems include diagrams of various 3D shapes like prisms, pyramids and cuboids. The document concludes with answers to all 10 problems.

1. PPR Maths nbk
MODULE 9
SKIM TUISYEN FELDA (STF) MATEMATIK SPM “ENRICHMENT”
TOPIC: LINES AND PLANES IN 3-DIMENSIONS
TIME: 2 HOURS
1 Diagram 2 shows a prism with cross section BCRQ. Given T and U are the
midpoint of AD and BC respectively, P and Q are right above T and U
respectively and PQRS is a square.
S 9 cm R
P Q 12 cm
D C
15 cm
T
U
DIAGRAM 2
A B
Calculate the angle between plane PBC and plane BCRQ.
[4 marks]
Answer :

2. PPR Maths nbk
2 Diagram 2 shows a cuboid with base TUVW.
P Q
5 cm 4 cm
T U
S R
W 12 cm V
DIAGRAM 2
Calculate the angle between plane PRV and plane QRVU.
[4 marks ]
Answer :

3. PPR Maths nbk
3
V
T
W
5 cm
U
S
12 cm
10 cm
R
DIAGRAM 3
Diagram 3 shows a right prism with horizontal rectangle base. Right triangle
RSW and UTV are the uniform cross section of the prism.
Calculate the angle between plane SRV and plane RSTU.
[4 marks]
Answer :

4. PPR Maths nbk
4 Diagram 4 shows a pyramid with a horizontal square base ABCD. T and U are
the midpoints of sides AB and CD respectively.
The height of the pyramid is 6 cm.
V
U
D • C
8cm
T
• DIAGRAM 4
A B
Calculate the angle between the line VT and the base ABCD.
[4 marks]
Answer :

5. PPR Maths nbk
5 Diagram 5 shows a cuboid PQRSDEFG with a horizontal square base PQRS.
D
E
J
P
G
Q F
S
R
DIAGRAM 5
J is the midpoint of DG. QR = RS = 12 cm and FR = 8cm.
Calculate the angle between the plane JRS and the plane RSGF.
[4 marks]
Answer :

6. PPR Maths nbk
6 Diagram 6 shows a right prism with an isosceles triangle base, STU. The
isosceles triangle STU is the uniform cross-section of the prism.
P Q
13 cm R
DIAGRAM 6
S T
•W 12 cm
10 cm
U
ST = SU and W is the midpoint of TU.
Calculate the angle between the line PW and the base STU.
[4 marks]
Answer :

7. PPR Maths nbk
7 Diagram 7 shows a right prism with a horizontal rectangular base PQRS.
VUQR is a trapezium. M and N are the midpoints of PS and QR
respectively.
Calculate the angle between the line TR and the base PQRS.
W [4 marks]
6 cm
T
V
S U
M 5 cm
•
P R
•
8 cm N
Q 12 cm
DIAGRAM 2

8. PPR Maths nbk
8 Diagram 8 shows a right prism. Right angled triangle PQR is the uniform cross-
section of the prism.
Q 18 cm R
9 cm
P
12 cm
T S
U
DIAGRAM 8
(a) Name the angle between the plane STP and the plane STQR,
(b) Calculate the angle between the plane STP and the plane STQR.
[4 marks]
Answer :
(a)
(b)

9. PPR Maths nbk
9 Diagram 9 shows a right prism with a horizontal rectangular base PQRS.
W
E
V
T
5 cm
U 7 cm
S
P
F
12 cm
R
Q 8 cm
DIAGRAM 9
Given that E and F are midpoints of WV and SR respectively.
(a) Find the length of PF
(b) Calculate the angle between the line PE and the plane PQRS
(c) Name the angle between the plane PQVE and the plane PQUT.
[6 marks]
Answer:
(a)
(b)
(c)

10. PPR Maths nbk
10 Diagram 10 shows a cuboid with a rectangular base PQRS.. M and N are
midpoints of TU and PQ respectively.
T M U
W V
7 cm
N
P Q
5 cm
S
24 cm R
DIAGRAM 10
(a) Calculate the length of SN
(b) Calculate the angle between the line SM and the plane TUVW
(c) Name the angle between the plane SRUM and the plane PQUT
[6 marks]
Answer :
(a)
(b)
(c)

11. PPR Maths nbk
MODULE 9 - ANSWERS
TOPIC: LINES AND PLANES IN 3-DIMENSIONS
1 Identify ∠PUQ @ ∠SCR
9
tan ∠PUQ =
12
∠PUQ = 360 52’ @ 36.870
2 Identify ∠ PRQ
12
tan ∠PRQ =
5
∠PRQ = 67.380 or 670 22’
3 Identify ∠VST
5
tan ∠VST =
12
22.60 or 220 37’
4 Identify ∠VTU
6
tan ∠VTU =
4
∠VTU = 56⋅31° or 56°18’
5 Identify ∠ GSJ
6
tan ∠ GSJ =
8
36.87° or 36° 52’
6 Identify ∠PWS
13
tan ∠PWS =
10 2 − 62
∠PWS = 58⋅39° or 58⋅4° or 58°24’
7 Identify ∠ TRM
Tan ∠ TRM = 5
8 + 62
2
∠ TRM = 26.57° or 26 ° 34 ′
8 Identify ∠PTQ or ∠QTP
9
tan ∠PTQ =
12

12. PPR Maths nbk
∠PTQ = 36052’ or 36.90
9 a) 10
b) Identify ∠EPF
7
Tan ∠EPF =
10
340 59’
c) ∠UQV
10 a) 13 cm
b) Identify ∠SMW
7
tan ∠SMW =
13
28.30 or 28018’
c) ∠RUQ or ∠QUR