Lines & Planes in 3D

8/3/2019 Lines & Planes in 3D

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Line And Planes In 3-Dimensions 1

CHAPTER 17 : LINES AND PLANES IN 3 DIMENSIONS

17.1 Angle Between Lines And Planes

Definitions

17.1 The angle between a line and a plane is the angle between the line and its orthogonal

projection on the plane.

1. 2.

O

C

AB

D

E

C

D

BA

E

Line DE and plane ABCD

Line AE and plane ABCD

3. 4.

CD

BA

E

CD

BA

E

Line CE and plane ABCD

Line BE and plane ABCD

P Q

RS

K

L

V

o

Line: KV

Plane : PQRSNormal to the plane: VL

Orthogonal projection of the lineonto the plane: KL

Angle between KV and the plane

PQRS is VKL


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17.1.1 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 : PQRSLine : KQ

Normal line :

Orthogonal projection :

Example 2 : Plane : PSK

Line : KR

Normal line : SROrthogonal projection : SK

2. a) Plane : CDEH

Line : GC

Normal line :Orthogonal projection :

b) Plane : BCV

Line : AV


Example 3 : Plane : GEVLine : VF

Normal line : FOOrthogonal projection : VO

3. a) Plane : CDVLine : BV


b) Plane : ADEFLine : DG


G H

EF

DA

B C

G H

EF

DA

B C

P Q

RS

LK

ED

FG

V

O A B

CD

V

A B

CD

V

G H

EF

DA

B C

DE

FG

V

O

P Q

RS

LK


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17.1.2 (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 : PQRS

Line : KQ

Angle :

Example 2 : Plane : PSKLine : KR

Angle : RKS

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

EF

DA

B C

G H

EF

DA

B C

P Q

RS

LK

ED

FG

V

OA B

CD

V

A B

CD

V

G H

EF

DA

B C

DE

FG

V

O

P Q

RS

LK


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Exercise 1 : Name the angle between the line and the plane given

a) Line UN and plane PQU b) Line JN 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 PQRS

J K

Q

M

RS

P

L

yN

y G

T

P Q

RS yN

y M

U

RS

U T

YX

BA

F

E

D C

G H

EF

LK

N M

P Q

RS

X

W

V

Ay

By

y L

y K


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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 KS 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 GFKN

f) The diagram shows a prism. Name the anglebetween line RV and the plane of PSWV.

J K

M

RS

P

L

R

S

U T

YX

BA

F

E

D C

G H

EF

LK

N M

P Q

RS

X

W

V

U

y Q

y P

DE

FG

V

O


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17.2 Angle Between Two Planes17.2 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 Draw 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.

LM and MN are perpendicular toDC.

The angle between the planeABCD and the plane CDEF is

KML ( or ADE or BCF )

M

L

NAB

CD

E F


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Finding Normal and Orthogonal Projection

1. 2.

Plane ABCD and plane CDEF

E

C

AB

D

F

PlaneABCD and plane CDEF

E

CD

BA

F

3. 4.

Plane ABCD and plane CDE

F

C

AB

E

PlaneABCD and plane CDG

J

H

G

F

CD

BA

E

5. 6.

Plane ABCD and plane ADE

F

C

AB

D

E

Plane ABCD and plane ADH

H

G

F

CD

BA

E


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17.2.1 a) Name the angle between the two planes.

Example 1: Plane EFGH andplane 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 PKS

2. a) Plane ABCD and plane

ADEF

b) Plane URST and plane

XRSY

Example 3 : Plane TRQ and

plane SRQP

Angle : TRS


ABV

b) Plane PQSR and plane

PQK

A B

CD

V

J K

Q

M

RS

P

L

P Q

R

X

W

U

G H

EF

DA

B C

BA

F

E

D C

RS

U T

YX

P Q

RS

LK

P Q

RS

LK

T

R

QP


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c) Plane JKLM and planePKL

d) Plane MHEL and planeNHE

e) Plane ABCD and planeBCE

Example 4: Plane DEV andDEFG

Angle : VMO

4a) Plane GCB and planeABCD

b) Plane KLMN and planeKPN

c) Plane ABE and plane

ABCD

d) Plane RUQ and plane

SRUT

e) Plane SURP and plane PTR

J K

Q

M

RS

P

L

BA

F

E

D C

G H

EF

LK

N M

D E

FG

V

O

y M A B

CD

G

O y L

BA

F

E

D C

y L

y K

R

S

T

Q

yN

y M

T

LM

NK

y E

y F

P

Q

y W

R

T

y V

S

P


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Exercise 1 : Name the angle between the two planes

a) Plane SRQP and plane QUTR b) Plane JQRM and plane JKLM

c) Plane RSYX and plane URST d) Plane BCF and plane ABCD

e) Plane GMLF and plane GHEF f) Plane PQA and plane PQRS

Cy

J K

Q

M

RS

P

L

RS

U T

YX

BA

F

E

D C

G H

EF

LK

N M

P Q

RS

X

W

V

Ay

By

T

Q

R

U

P

S


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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 ofDEFG.

b) The diagram shows a cuboid with a

horizontal base JKLM .Name the angle

between the plane SRKJ and the plane ofSRLM.

c) The diagram shows a prism. Name theangle between the plane RSY and the plane of

RSTU.

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 anglebetween the plane UVWX and the plane of

PSWV.

J K

M

RS

P

L

RS

U T

YX

BA

F

E

D C

G H

EF

LK

NM

y Q

y P


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Questions According To SPM Format

1. Diagram 1 shows a pyramid with a

rectangular base PQRS. V is vertically above P.

Name the angle between the line VR and theplane PQRS

A. PRV B. VRS

C. PVQ D. VQS

2. Diagram 2 shows a cuboid.

Name the angle between the line PM and theplane SRMN.

A. PMK B. PMQ

C. PMR D. PMS

3. In the diagram 3, X and Y are the midpoints

of PW and SV respectively.

The angle between line RX and the plane

RSVU 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. EKF

C. HEK D. JEK

K L

MN

RS

P Q

P Q

RS

V

DIAGRAM 1 DIAGRAM 2

Q P

WS

VU

R T

y X

DIAGRAM 3

y Y

F G

K

H

E J

DIAGRAM 4


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5. Diagram 5 shows a pyramid .PQRS is a

rectangle.

Name the angle between the line TQ and theplane PQRS

A. QPT B. QST

C. RST D. SQT


The angle between the line SU and planePSWT is

A. USP B. USQ

C. UST D. USW


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 andthe plane FGKJ.

A. EGF B. EKF

C. HGJ D. GKE

T U

VW

RS

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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9. Diagram 9 shows a right prism .

Name the angle between the plane ACS and theplane SADR.

A. CAS B. SCD

C. ACD D. CAD

10. Diagram 10 shows a right pyramid

MABCD 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 an

equilateral triangle and PV = VR = SW = WU .

The angle between the plane PRT and the plane

PRUS 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 planeJKLM.

A. YBA B. AYB

C. YAB D. ABY

DIAGRAM 11

DIAGRAM 9

Q C

DR

AS

P B

Q

P

y WS

U

R

T

y V

J K

L

Y

X

M

DIAGRAM 12

y A

y B

DIAGRAM 10

A B

CD

O


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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 triangle

RUQ and STP are the uniform cross-section ofthe prism. M and N are the mid-points of STand RU, respectively.

The angle between the plane PRN and the

plane RSTU is

A. PRS B. MRP

C. PNM D. NPM

16. Diagram 16 shows a cuboid with base

PQRS . .

Name the angle between the plane WXR and

the plane QRWV.

A. QRX B. RXQ

C. RWX D. SXU

DIAGRAM 14DIAGRAM 13

C

D

y MA

B

F

L

JK

HG

R

U

S

P

T

Q

DIAGRAM 15

yN

y MR

Q

U

W

T

SP

V

y X

DIAGRAM 16


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17. Diagram 17 shows a cuboid with

rectangular 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 a

horizontal plane EFGH. J is a mid-point ofGH.

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 horizontalplane QRP and vertical rectangular QRST .

Name the angle between the plane PRS and the

plane QRST is

A. PSQ B. PST

C. PRT D. PRQ

20. Diagram 20 shows a prism withrectangular base JKLM . LM is normal to baseJKLM .

Name the angle between line NK and base

JKLM.

A. NKM 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

LM


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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 andN 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. PNQ D. MPN

2. SPM July 2004, Q16

Diagram 9 shows a cuboid with PQRS as its horizontal base.

Name the angle between the plane TQR and the plane TUVW.

A. TRW B. TQV

C. WTR D. VTQ

3. SPM Nov 2004, Q14

T

R

US y M

yN

P

P

Q

RS

WT

U V

DIAGRAM 6

DIAGRAM 9


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Diagram 8 shows a cuboid with horizontal base PQRS.

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 NPQRS with square base PQRS.

The angle between the line NQ and the base PQRS is

A. PQN B. NQS

C. QNS D. QNR

5. SPM Nov 2005, Q14

P

RS

WT

U

Q R

SP

DIAGRAM 8

DIAGRAM 6


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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

SP

Q R

DIAGRAM 7


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7. SPM Nov 2006, Q14


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 PQRS. 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 RPS

B RPQ

C PRS

D PRQ

P

Q

RS

Diagram


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9. SPM Nov 2007 Q14

Diagram 8 shows a pyramid with its rectangle base QRST.

Vertex P is vertically above T.

Name the angle between the plane PTS and the plane PTQ.

A PQT

B PST

C SPQ

D STQ

10. SPM Jun 2008, Q 15.


What is the angle between the plane SVW and the plane PQRS?

A QSW

B RSW

C QSV

D PSV

P

Q

R

S

T

U

V

W

Diagram 8

P

Q

R

ST

Diagram 8


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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 USV

D SUV

P

Q S

T

U

V


23/41


ANSWERS

Exercise 1a) NUM b) NJG

c) SXT d) EBL

e) KMF f) APB

Exercise 2

a) VGO b) LSKc) RYT d) EQP

e) ENF f) RVS

17.2.4

1a) JKP or MLSb) UVP or XWS

2a) BAF or CDEb) URX or TSY

3a) VBC b) KPS c) PKJd) NHM e) ECD

4a) GLO b) PEF c) ELK

d) QNM e) TVW

Exercise 1

a) PQU or SRT

b) QJK or RML

c) XRU or YSTd) FBA e) MGH or LFE

f) ACB

Exercise 2

a) VLO b) MSJ or KRLc) YST d) EQP e) KEFf) 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. B11. 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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Problems Solving

The Angle Between The Line And The PlaneExercise 1 : Based on the diagram, calculate the angle between the line and the plane given

a) The diagram shows a cuboid. Calculate theangle between line NE and the plane of GFKN

b) The diagram shows a cuboid with ahorizontal base JKLM .Calculate the angle

between 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

M

RS

L

6 cm

5 cm

cm

RS

U T

YX

cm

8 cm

14 cm

BA

F

E

D C

y Q

y P

6 cm

5 cm

12 cm

G H

EF

LK

N M12 cm

5 cm

16 cm


25/41


e) The diagram shows a pyramid . Given that

HP = 13 cm. Calculate the angle between linePG and the plane of EHP.

f) The diagram shows a prism. Calculate the

angle 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

5 cm

4 cm

3 cm

7 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 and

plane 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

P

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


27/41


Example 3 : Plane TRQ and

plane SRQP

Angle : TRS

tan TRS =RS

TS

=11

4

QLR = 19.98o / 19o59


ABV

b) Plane PQSR and plane

PQKL

Example 4: Plane DEV andDEFG . VO = 7 cm

Angle : VMO

tan VMO =MO

VO

=6

7

VMO = 49.40o / 49o24

4a) Plane GCB and planeABCD

b) Plane PMNT and PlaneKLMN

A B

C

D

V

P Q

RS

LK

T

R

QP

AB

CD

G

O y L

5 cm

5 cm

11 cm

4 cm

8 cm

5 cm4 cm

3 cm

DE

FG

V

Oy M

10 cm

12 cm8 cm

12 cm

10 cm

T

L M

K

yF

P

9 cm

12 cm

10 cm


28/41


Example 5 : Plane ABE and

plane ABCD

Angle : ELK

EK =22

915 = 12

tan ELK =LK

EK

=36

12

ELK =

4 a) Plane SRQ and plane

SRUT

b) Plane SURP and plane PTR

Exercise 1

a) The diagram shows a pyramid with ahorizontal 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 ahorizontal base JKLM .Calculate the angle

between the plane SRKJ and the plane of

SRLM.

BA

F

E

D

C

y L

y K

18 cm

15 cm

36 cm

R

S T

Q

yN

y M

8 cm

5 cm

P

y WU

R

T

y V

10 cm

4 cm

12 cm

J K

M

RS

L

BC

DA

V

O

10 cm

8 cm

7 cm

6 cm9 cm


29/41


c) The diagram shows a prism. Calculate the

angle between the plane PLM and the plane ofPLNQ.

d) The diagram shows a prism. Calculate the

angle 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

y A

y 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


30/41


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 cm

vertically above the point J. Calculate the anglebetween 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 y

Q R

SP

L

10 cm

12 cm

J y

Q R

SP

L

10 cm

12 cm

J y

Q R

SP

L

10 cm

12 cm

J y


31/41


Example 2

Diagram 2 shows a prism with horizontal

square ABCD. Trapezium KABL is the

uniform cross-section of the prism. Therectangular 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 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 / 36

o52

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


32/41


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 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 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 = 4

cm.

Calculate the angle between the plane ABH

and 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


33/41


5) Diagram 5 shows a pyramid with the

horizontal 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 the

midpoint of TW .

Calculate the angle between plane YVZ and thehorizontal plane XYVW.

7) Diagram 7 shows a right prism with basethe rectangular plane ABCD. Right triangle

BCF 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. Thebase 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 theplane EFGH.

Y V

WX

TS

RU

10 cm

6 cm

4 cm

y Z

DIAGRAM 6

T

R

QP

12 cm13 cm

10 cm

DIAGRAM 5

B

DIAGRAM 7 DIAGRAM 8

EF

GH

R

5 cm

24 cm

y S

A

CD

FE

8 cm

6 cm

6 cm


34/41


9) Diagram 9 shows a cuboid. P is the midpoint

of line RQ.

Calculate the angle between the plane LQY andthe plane MQRN.

10) Diagram 10 shows a right prism. Right

angled 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 the

plane SRML.

12) Diagram 12 shows a right prism with

rectangle base EFGH. EFPQ and GHPQ arerectangle.

Calculate the angle between line LQ and the

base EFGH.

DIAGRAM 9

L M

QP

RS

K N y Y

10 cm

6 cm

12 cm

U

Q

ST

P

R

5 cm12 cm

20 cm

DIAGRAM 10

P Q

RS

ML

y X

12 cm

8 cm

5 cm

DIAGRAM 11

F

G

E

P

H

y M

y L

6 cm

5 cm

12 cm

DIAGRAM 12


35/41


Past Year SPM Questions

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 surface

GFLH 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

HJE

L

6 cm

8 cm

Dia ram 1

A B

G

D C

E

F

H

12 cm

5 cm

cm

DIAGRAM 2


36/41


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 isinclined.

Calculate the angle between the plane ABE and the base ABCD. [3 marks]

B

E

A

CD

F

12

5 cm

3

DIA RAM

DIAGRAM 2

K J

ML

V

10 cm

12

Q y


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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

5

1

DIAGRAM 1

DIAGRAM 2

J

M

H

KL

N

6 cm

12


38/41


7. Nov 2006, Q2

Diagram 1 shows a right prism. The base PQRS is on horizontal rectangle. The right triangleUPQ 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


39/41


9. SPM Nov 2007 Q4Diagram 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 2008Diagram 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

68

5


40/41


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


41/41

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 /38

o40

3e 18.43o / 18o26

Exercise 1a 50.91

o /

50o54

b 26.57o / 26

o34 c 54.46o / 54

o28 d 71.57o /

71o34

e 28.30o /28o18

f 51.34o / 51o20 g 54.16o / 54o10 h 53.13o /53o8

9.2.11a 36.87

o /36

o52

1b 74.05o / 74o3 2a 67.38

o / 67o23 2b 29.05

o /29

o3

3a 57.99o / 58o 3b 36.89o / 36o52 4a 60o 4b 53.13o /53o8

5a 36.87o

/36

o52

5b 63.43o

/ 63o26

Exercise 1

a 66.04o /66

o2 b 33.69o / 33o41 c 26.57o /

26o34d 66.42o /

66o25

e 37.87o

/37o52

f 34.70o

/ 34o42

PRACTICE SPM FORMAT

1 47.73o

/47o44

2 17.10o

/ 17o6 3 54.46

o/

54o284 56.31

o/

56o19

5 63.43o /

63o26

6 36.87o / 36

o52 7 36.87

o /

36o52

8 34.70o /

34o429 30.96

o /

30o58

10 30.96o / 30o58 11 53.13

o /

53o8

12 18.43o /

18o26SPM PAST YEAR QUESTIONS

1 Nov 2003 36.87o / 36o52

2 Jul 2004 18.43o / 18

o 26

3 Nov 2004 31.61o / 31o 36

4 Jul 2005 14.04o

/ 14o

2

5 Nov 2005 33.69o / 33o 41

6 Jul 2006 50.19o

/ 50o

127 Nov 2006 34.70 / 34

O42

8 Jun 2007 ,54.46 or54 28'WQR r r

9 Nov 2007 SUV , 31.61 or 31 36'r r

10 Jun 2008 ,32QPR r

11 Nov 2008 ,15.47 or 15 28'EAM r r

Lines & Planes in 3D

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