chapter 9 ii lines & planes in 3d enhance

8/3/2019 Chapter 9 II Lines & Planes in 3D ENHANCE

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CHAPTER 9 : 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 / 26o 34

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 / 67o 23

2. a) Plane : CDEH

Line : FD

Angle :

b) Plane : URST

Line : RX

Angle :

Example 3 : Plane : JKLM 3. a) Plane : ABCD b) Plane : ABCD

Lines and Planes in 3-Dimensions 1

PQ

RS

LK

G H

EF

DA

B C

6 cm

8 cm

4 cm

PQ

RS

LK

12 cm

7 cm

5 cm

AB

CD

V

10 cm

8 cm

3 cm

12 cm

5 cm

GH

EF

DA

B C

15 cm

6 cm

8 cm

R S

UT

YX

24 cm

7 cm

4 cm


2/21

Line : NK

NM = 11 cm

Angle : NKM

KM = 22 912 + = 15 cm

tan NKM =KM

NM

=15

11

NKM = 36.25o / 36o 15

Line : AV

Line : DG

c) Plane : QRSTLine : TP

d) Plane : QPWTLine : RX

e) Plane : SRUTLine : PN

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


J K

LM

N

12 cm

9 cm

AB

CD

V

8 cm

6 cm

4 cm

G H

EF

DA

B C

6 cm

5 cm

12 cm

5 cm R

T

Q

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

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 angle

between line KS and the plane of SRLM.

c) The diagram shows a prism. Calculate the

angle between line RY and the plane of STY.

d) The diagram shows a prism. Calculate the

angle between line QE and the plane of DCE.

e) The diagram shows a pyramid . Given that

HP = 13 cm. Calculate the angle between line

f) The diagram shows a prism. Calculate the

angle between line UV and the plane of PSWV.


J K

Q

M

RS

P

L

6 cm

5 cm

8 cm

R S

UT

YX

6 cm

8 cm

14 cm

BA

F

E

D C

Q

P

6 cm

5 cm

12 cm

GH

EF

LK

N M

16 cm

5 cm

12 cm


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PG and the plane of EHP.

g) The diagram shows a pyramid with a

horizontal base DEFG. Given that VO = 9 cm.

Calculate the angle between line GV and the

plane of DEFG.

h) The diagram shows a pyramid with a

triangle base CHD. Calculate the angle

between line CA and the plane of ADH.

9. 2 Angle Between Two Planes


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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9.2.1 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 / 56o19

1. a) Plane KLSP and plane

JKLM

b) Plane PSWV and plane

VUXW

Example 2 : Plane PQLK and

plane SRLK

Angle :

QLR = PKS

tan QLR =LR

QR

=710

QLR = 55o

2. a) Plane ABCD and plane

ADEF

b) Plane URST and plane

XRSY

Example 3 : Plane TRQ and

plane SRQP

3. a) Plane ABCD and plane

ABV

b) Plane PQSR and plane

PQKL


AB

CD

V

GH

EF

DA

B C

PQ

RS

LK

T

RS

QP

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

PQ

RS

LK

12 cm

10 cm

7 cm

BA

F

E

D C

20 cm

10 cm

13 cm

5 cm

11 cm

4 cm

3 cm

R S

UT

YX

12

cm

9 cm

5 cm


6/21

Angle : TRS

tan TRS =RS

TS

=11

4

QLR = 19.98o / 19o59

Example 4: Plane DEV and

DEFG . VO = 7 cm

Angle : VMO

tan VMO =MO

VO

=6

7

VMO = 49.40o / 49o24

4a) Plane GCB and plane

ABCD

b) Plane PMNT and Plane

KLMN


AB

CD

G

OL

5 cm

4 cm

8 cm

5 cm

DE

FG

V

OM

10 cm

12 cm

8 cm

12 cm

10 cm

T

L M

NK

F

P

9 cm

12 cm

10 cm


7/21

Example 5 : Plane ABE and

plane ABCD

Angle : ELK

EK = 22 915 = 12

tan ELK =LK

EK

=3612

ELK =

4 a) Plane SRQ and plane

SRUT

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 angle

between the plane SRKJ and the plane of

SRLM.

c) The diagram shows a prism. Calculate the d) The diagram shows a prism. Calculate the


BA

F

E

D C

L

K

18 cm

15 cm

36 cm

R

U

S

P

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

P

L

BC

DA

V

O

10 cm

8 cm

7 cm

6 cm9 cm

QP

D

C

S R

A

B

8 cm

10 cm

15 cm


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angle between the plane PLM and the plane of

PLNQ.

angle between the plane QRC and the plane of

PQRS.

e) The diagram shows a pyramid. Calculate

the angle between the plane FGP and the plane

of EFGH

f) The diagram shows a prism. Name the angle

between the plane ABCD and the plane of

DQR.

How to answer the SPM format Question


K M

LN

QP

20 cm

10 cm

5 cm

E F

GH

P

18 cm

24 cm

14cm

PQ

RS

CD

A B

13 cm

7cm

9cm


9/21

Example 1

Diagram 1 shows a pyramid LPQRS .

The base PQRS is a horizontal rectangle. J is

the 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 : 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

cm

Q R

SP

L

10 cm

12 cm

J

cm

Q R

SP

L

10 cm

12 cm

J

cm

Q R

SP

L

10 cm

12 cm

J

cm


10/21

Example 2

Diagram 2 shows a prism with horizontal

square ABCD. Trapezium KABL is the

uniform cross-section of the prism. The

rectangular 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 plane

NBC 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

Questions Based on the Examination Format


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


11/21

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 and

the plane FEHG.

4. Diagram 4 shows a right prism with a

horizontal plane ABCD. It is a uniform prism

and its cross section is an isosceles triangle of

sides 4 cm. The thickness of the prism, EA = 4

cm.

Calculate the angle between the plane ABH and

the plane ABE.

5) Diagram 5 shows a pyramid with the 6) Diagram 6 shows a cuboid. Z is the


DIAGRAM 1

DIAGRAM 2

DIAGRAM 3

A B

C

H

E D

DIAGRAM 4

KL

MN

RS

P Q

12 cm

4 cm

5 cm

P Q

RS

V

8 cm

6 cm

11 cm

F E

HB

CD

A

G

7 cm

5 cm

6 cm

4 cm

YV

WX

TS

R U

10 cm

6 cm

4 cm

Z

DIAGRAM 6


12/21

horizontal plane, TRS. The rectangle PQRS is

vertical plane.

Calculate the angle between the plane PTS and

the plane TQR.

midpoint 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 triangle

BCF is the uniform cross-section of the prism.

The rectangular surface DCFE is vertical while

the rectangular surface BAEF is inclined.

Calculate the angle between the plane DB and

plane EDCF.

8) Diagram 8 shows a pyramid REFGH. The

base 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 the

plane EFGH.

9) Diagram 9 shows a cuboid. P is the midpoint 10) Diagram 10 shows a right prism. Right


T

RS

QP

12 cm

13 cm

10 cm

DIAGRAM 5

B

DIAGRAM 7 DIAGRAM 8

EF

GH

R

5 cm

24 cm

S

LM

QP

RS

KN

Y

10 cm

6 cm

12 cm

A

CD

FE

8 cm

6 cm

6 cm


13/21

of line RQ.

Calculate the angle between the plane LQY and

the plane MQRN.

angled triangle SUT is the uniform cross-

section of the prism.

Calvulate the angle between the plane PSR and

the plane PUTR..

11) Diagram 11 shows a prism . The base

PQRS is a horizontal rectangle . X is the

midpoint 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 are

rectangle.

Calculate the angle between line LQ and the

base EFGH.

Past Year SPM Questions


DIAGRAM 9 cm

U

Q

ST

P

R

5 cm12 cm

20 cm

DIAGRAM 10

PQ

RS

ML

X

12 cm

8 cm

5 cm

DIAGRAM 11 cm

F

G

E

P

H

Q

M

L

6 cm

5 cm

12 cm

DIAGRAM 12 cm


14/21

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 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


15/21

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 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

cm


16/21

5. Nov 2005, Q4

Diagram 1 shows a right prism. Right angled triangled PQR is the uniform cross-section of

the 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 angled

triangle NHJ is the uniform cross-section of the prism.

Identify and calculate the angle between the line KN and the plane HLMN.

7. Nov 2006, Q2


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

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. TrapeziumPQVU is the uniform cross-section of the prism. The rectangle QRWV is a vertical plane

and 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 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 2008

Diagram shows a cuboid ABCDEFGH with horizontal base ABCD. P, Q and R are themidpoints 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

9.1.1

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 /

38o40

3e 18.43o / 18o26

Exercise 1

a 50.91o /50o54

b 26.57o / 26o34 c 54.46o / 54o28 d 71.57o /71o34

e 28.30o /

28o18

f 51.34o / 51o20 g 54.16o / 54o10 h 53.13o /

53o8

9.2.1

1a 36.87o /

36o52

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

29o3

3a 57.99o / 58o 3b 36.89o / 36o52 4a 60o 4b 53.13o /

53o8

5a 36.87o /

36o52

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 / 36o52 7 36.87o /

36o52

8 34.70o /

34o42

9 30.96o /

30o58

10 30.96o / 30o58 11 53.13o /

53o8

12 18.43o /

18o26



21/21

SPM PAST YEAR QUESTIONS

1 Nov 2003 36.87o / 36o 52

2 Jul 2004 18.43o / 18o 26

3 Nov 2004 31.61o

/ 31

o

364 Jul 2005 14.04o / 14o 2

5 Nov 2005 33.69o / 33o 41

6 Jul 2006 50.19o / 50o 12

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

chapter 9 ii lines & planes in 3d enhance

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