[Worksheet] Lines & Planes in 3D (1)

7/27/2019 [Worksheet] Lines & Planes in 3D (1)

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Lines and Planes in 3-Dimensions 1

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/ 26o34

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 / 67o23

2. a) Plane : CDEH

Line : FD

Angle :

b) Plane : URST

Line : RX

Angle :

PQ

RS

LK

G H

EF

DA

B C

6 cm

8 cm

4 cm

P Q

RS

LK

12 cm

7 cm

5 cm

A B

CD

V

10 cm

8 cm

3 cm

12 cm

5 cm

G H

EF

DA

B C

15 cm

6 cm

8 cm

RS

U T

YX

24 cm

7 cm

4 cm


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Example 3 : Plane : JKLMLine : NK

NM = 11 cm

Angle : NKM

KM =22 912 + = 15 cm

tan NKM =KM

NM

=15

11

NKM = 36.25o/ 36o15

3. a) Plane : ABCDLine : AV

b) Plane : ABCDLine : DG

c) Plane : QRST

Line : TP

d) Plane : QPWT

Line : RX

e) Plane : SRUT

Line : PN

J K

LM

N

12 cm

9 cm

A B

CD

V

8 cm

6 cm

4 cm

G H

EF

DA

B C

6 cm

5 cm

12 cm

5 cm R

T

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


3/21


Exercise 1 : Based on the diagram, calculate the angle between the line and the plane given

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 anglebetween 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

Q

M

RS

L

6 cm

5 cm

8 cm

RS

U T

YX

6 cm

8 cm

14 cm

BA

F

E

D C

Q

P

6 cm

5 cm

12 cm

G H

EF

LK

N M12 cm

5 cm

16 cm


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e) The diagram shows a pyramid . Given thatHP = 13 cm. Calculate the angle between linePG and the plane of EHP.

f) The diagram shows a prism. Calculate theangle 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

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 Angle Between Two Planes

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/ 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

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

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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Example 3 : Plane TRQ andplane SRQP

Angle : TRS

tan TRS =RS

TS

=114

QLR = 19.98o/ 19o59

3. a) Plane ABCD and planeABV

b) Plane PQSR and planePQKL

Example 4: Plane DEV and

DEFG . VO = 7 cm

Angle : VMO

tan VMO =MO

VO

=67

VMO = 49.40o/ 49o24

4a) Plane GCB and plane

ABCD

b) Plane PMNT and Plane

KLMN

A B

CD

V

P Q

RS

LK

T

RS

QP

AB

CD

G

O L

5 cm

5 cm

11 cm

4 cm

8 cm

5 cm4 cm

3 cm

DE

FG

V

O

M10 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 andplane ABCD

Angle : ELK

EK =22 915 = 12

tan ELK =LK

EK

=36

12

ELK =

4 a) Plane SRQ and planeSRUT

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 anglebetween the plane SRKJ and the plane of

SRLM.

BA

F

E

D C

L

K

18 cm

15 cm

36 cm

R

U

S 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

L

BC

DA

V

O

10 cm

8 cm

7 cm

6 cm9 cm


8/21


c) The diagram shows a prism. Calculate theangle between the plane PLM and the plane ofPLNQ.

d) The diagram shows a prism. Calculate theangle 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

A

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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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 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 : LJOrthogonal projection : QJ

Step 3 :

Identify the angle

Angle : LQJ

Step 4 :

Calculate the angle

JQ =22 512 = 13

tan LQJ =QJLJ

=13

8

LQJ =

Q R

SP

L

10 cm

12 cm

Diagram 1

J

Q R

SP

L

10 cm

12 cm

J

Q R

SP

L

10 cm

12 cm

J

Q R

SP

L

10 cm

12 cm

J


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Example 2Diagram 2 shows a prism with horizontal

square ABCD. Trapezium KABL is theuniform cross-section of the prism. Therectangular surface NKAD is vertical while the

rectangular surface MLBC is inclined.

Calculate the angle between the plane NBC andthe 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/ 36o52

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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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 andthe plane PQRS.

2. Diagram 2 shows a cuboid with horizontal

base KLMN.

Calculate the angle between the line SL and thebase 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 = 4cm.

Calculate the angle between the plane ABHand 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


12/21


5) Diagram 5 shows a pyramid with thehorizontal 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 themidpoint 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 triangleBCF 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. The

base EFGH is a horizontal rectangle. R is themidpoint 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

T

S

R U

10 cm

6 cm

4 cm

Z

DIAGRAM 6

T

RS

QP

12 cm

13 cm

10 cm

DIAGRAM 5

B

DIAGRAM 7 DIAGRAM 8

E

GH

R

5 cm

24 cm

S

A

CD

FE

8 cm

6 cm

6 cm


13/21


9) Diagram 9 shows a cuboid. P is the midpointof line RQ.

Calculate the angle between the plane LQY andthe plane MQRN.

10) Diagram 10 shows a right prism. Rightangled 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 theplane SRML.

12) Diagram 12 shows a right prism with

rectangle base EFGH. EFPQ and GHPQ arerectangle.

Calculate the angle between line LQ and thebase EFGH.

DIAGRAM 9

L M

QP

RS

K N Y

10 cm

6 cm

12 cm

U

Q

ST

P

R

5 cm12 cm

20 cm

DIAGRAM 10

P Q

RS

ML

X

12 cm

8 cm

5 cm

DIAGRAM 11

F

G

E

P

H

M

L

6 cm

5 cm

12 cm

DIAGRAM 12


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Past Year SPM Questions

1. Nov 2003

Diagram 1 shows a prism with a horizontal square base HJKL. Trapezium EFLK is the uniformcross-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]

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 angledtriangle 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]

B

E

A

CD

F

12 cm

5 cm

3 cm

DIAGRAM 1

DIAGRAM 2

K J

ML

V

10 cm

12 cm

Q


16/21


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 cm

5 cm

18 cm

DIAGRAM 1

DIAGRAM 2

J

M

H

KL

N

6 cm

12 cm

8 cm


17/21


7. Nov 2006, Q2

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. 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


18/21


9. SPM Nov 2007 Q4

Diagram shows a right prism. The base PQRS is a horizontal rectangle. Right angled

triangle 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 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

Q

6 cm8 cm

5 cm


19/21


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


20/21


ANSWERSChapter 9 :Lines And Planes In 3 Dimensions

9.1.11a 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/

38o403e 18.43

o/ 18

o26

Exercise 1a 50.91o/

50o54

b 26.57o/ 26o34 c 54.46o/ 54o28 d 71.57o/

71o34e 28.30o/

28o18f 51.34o/ 51o20 g 54.16o/ 54o10 h 53.13o/

53o8

9.2.11a 36.87o/

36o52

1b 74.05o/ 74o3 2a 67.38o/ 67o23 2b 29.05o/

29o3

3a 57.99o/ 58

o 3b 36.89

o/ 36

o52 4a 60

o 4b 53.13

o/

53o8

5a 36.87o/36

o52

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/ 36

o52 7 36.87

o/

36o52

8 34.70o/

34o42

9 30.96o/30

o58

10 30.96o/ 30o58 11 53.13o/53

o8

12 18.43o/18

o26


21/21


SPM PAST YEAR QUESTIONS

1 Nov 2003 36.87o/ 36o522 Jul 2004 18.43

o/ 18o26

3 Nov 2004 31.61o/ 31

o36

4 Jul 2005 14.04o/ 14o2

5 Nov 2005 33.69o/ 33

o41

6 Jul 2006 50.19o/ 50o12

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

Documents

[Worksheet] Lines & Planes in 3D (1)