chapter 17 i lines & planes in 3d enhance

8/3/2019 Chapter 17 I Lines & Planes in 3D ENHANCE

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Line And Planes In 3-Dimensions 1

CHAPTER 17 : LINES AND PLANES IN 3 DIMENSIONS

Definitions

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

projection on the plane.

11.2 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 D

raw 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.

P Q

R S

K

L

V

o

Line: KV

Plane : PQR S

Normal to the plane: VLOrthogonal projection of the line

onto the plane: KL

Angle between KV and the plane

PQR S is VK L

LM andMN are perpendicular toDC.

The angle between the plane

ABCD and the plane CDEF is

K ML ( or ADE or BCF )

M

L

N A B

C D

E F




17.1 Angle Between Lines And Planes

17.1.1 Draw and /or colour the given line

Example 1 : Line AH 1. a) Line BE b) Line FD

Example 2: Line PL 2.a) Line QS b) Line RK

Example 3: Line VF 3. a) Line VM b) Line V N

Example 4 : Line DV 4. a) Line AC b) Line BV

G H

E F

D A

B C

G H

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

P Q

R S

LK

P Q

R S

LK

D E

F G

V

O

D E

F G

V

O

D E

F G

V

OMy

Ny

A B

C D

V

A B

C D

V

A B

C D

V




17.1.2 Shade Or Colour The Plane Given

Example 1: Plane EFGH 1. a) Plane DEFG b) Plane PQR S

Example 2 : Plane PSK 2. a) Plane ABGF b) Line BCV

Example 3 : Plane DEV 3. a) Plane ADHG b) Plane ADV

c) Plane ACV d) Plane VGO e) Line K SQ

G H

E F

D A

B C

G H

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

P Q

R S

LK

P Q

R S

LK

D E

F G

V

O

D E

F G

V

O

D E

F G

V

O

A B

C D

V

A B

C D

V

A B

C D

V




17.1.3 Complete the following question

i) shade the plane givenii) draw the line given

iii) determine the point which lies on the plane given

Example 1: Plane :EFGH

Line :GD

1. a) Plane : DEFGLine : DV

b) Plane : PQR S Line : KQ

Example 2 : Plane : PSK Line : KR

2. a) Plane : CDEHLine : GC

b) Plane : BCVLine : AV

Example 3 : Plane : GEV

Line : VF

3. a) Plane : CDV

Line : BV

b) Plane : ADEF

Line : DG

c) Plane : ABCD

Line : AV

d) Plane : DVF

Line : EV

e) Plane : K SP

Line : KQ

G H

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

P Q

R S

LK

E D

F G

V

O

D E

F G

V

OA B

C D

V

A B

C D

V

A B

C D

V

G H

E F

D A

B C

D E

F G

V

O

P Q

R S

LK




17.1.4 Name and draw the i) normal line

ii) orthogonal projection


Line :GD

Normal line : DE

Orthogonal projection : GE

1. a) Plane : DEFGLine : DV

Normal line :

Orthogonal projection :

b) Plane : PQR S Line : KQ

Normal line :

Orthogonal projection :

Example 2 : Plane : PSK

Line : KR

Normal line : SR Orthogonal 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 : ADEF Line : DG


G H

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

E D

F G

V

O A B

C D

V

A B

C D

V

G H

E F

D A

B C

D E

F G

V

O

P Q

R S

LK




17.1.5 (a) Name the angle between the line and the plane given


Line :GD

Angle : DGE

1. a) Plane : DEFG

Line : DV

Angle :

b) Plane : PQR S

Line : KQ

Angle :

Example 2 : Plane : PSK Line : KR

Angle : RK S

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

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

E D

F G

V

OA B

C D

V

A B

C D

V

G H

E F

D A

B C

D E

F G

V

O

P Q

R S

LK




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

a) Line U N and plane PQU b) Line J N 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 PQR S

J K

Q

M

R S

P

L

y N

yG

T

P Q

R S y N

yM

U

R S

U T

YX

BA

F

E

D C

G H

E F

LK

N M

P Q

R S

X

W

V

Ay

By

yL

yK




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 K S 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 GFK N

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

J K

M

R S

P

L

R

S

U T

YX

BA

F

E

D C

G H

E F

LK

N M

P Q

R S

X

W

V

U

yQ

y P

D E

F G

V

O




17. 2 Angle Between Two Planes

17.2.1 Shade or colour the planes given

Example 1 :

Plane GHEF and plane BHEA

1. a) Plane ABCD and plane

ABHE

b) Plane AGHD and plane

ABCD

Example 2:Plane PLR and QRL

2.a) Plane PSK and planeK SQ

b) PlaneSRLK and planeKLP

Example 3:

Plane VEF and plane DEFG

3. a) Plane DVG and plane

VEF

b) Plane VDE and plane

DEFG

Example 4 :

Plane ABV and plane ABCD

4. a) Plane AVC and plane

VBC

b) Plane VDC and plane VBC

G H

E F

D A

B C

G H

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

P Q

R S

LK

P Q

R S

LK

D E

F G

V

OD

E

F G

V

O

D E

F G

V

O

Ny

A B

C D

V

A B

C D

V

A B

C D

V




17.2.1 Name the line of intersection between the two planes

Example 1 :

Plane GHEF and plane BHEA

Line : HE


ABHE

Line : «««..

b) Plane AGHD and plane

ABCD

Line : «««

Example 2:

Plane PLR and QRL

Line : RL

2.a) Plane PSK and plane

K SQ

Line : ««««

b) PlaneSRLK and plane

KLP

Line : ««««

Example 3:

Plane VEF and plane DEFG

Line : EF

3. a) Plane DVG and plane

DEFG

Line : ««««

b) Line VDE and plane

DEFG

Line : ««««

Example 4 :

Plane ABV and plane ABCD

Line : AB

4. a) Plane DAV and plane

DAC

Line : ««««

b) Plane VDC and plane VBC

Line : ««««

G H

E F

D A

B C

G H

E F

D A

B C

G H

E F

D A

B C

P Q

R S

LK

P Q

R S

LK

D E

F G

V

O

D E

F G

V

O

Ny

A B

C D

V

A B

C D

V

P Q

R S

LK

D E

F G

V

O

A B

C D

V




17.2.2 Name and colour the perpendicular line to the line intersection between the two planes

Example 1: Plane EFGH and plane GHDA

Line EH and DH or line

FG and AG are

perpendicular to GH

1. a) Plane KLSP and plane

JKLM

b) Plane PSWV and plane

VUXW

Example 2 : Plane PQLK and

plane SRLK

Line PK and SK or line

QL and RL are

perpendicular to K L


ADEF

b) Plane UR ST and plane

XR SY

Example 3 : Plane TRQ and plane SRQP

Line TR and SR is

perpendicular to QR

3. a) Plane ABCD and planeABV

b) Plane PQSR and planePQK

A B

C D

V

J K

Q

M

R S

P

L

P Q

R

X

W

U

G H

E F

D A

B C

BA

F

E

D C

R S

U T

YX

P Q

R S

LK

P Q

R S

LK

T

R

QP




c) Plane JKLM and plane

PKL

d) Plane MHEL and plane

NHE

e) Plane ABCD and plane

BCE

Example 4: Plane DEV and

DEFG

Line VM and OM is

perpendicular to DE

4a) Plane GCB and plane

ABCD

b) Plane KLMN and plane

KP N

c) Plane ABE and plane

ABCD

d) Plane RUQ and plane

SRUT

e) Plane SURP and plane PTR

J K

Q

M

R S

P

L

BA

F

E

D C

G H

E F

LK

N M

D E

F G

V

O

yM A B

C D

G

O yLT

LM

N K

yE

yF

P

BA

F

E

D C

yL

yK

R

S

T

Q

y N

yM

P

S

Q

yWU

R

T

yV




17.2.3 a) Name the angle between the two planes.

Example 1: Plane EFGH and plane 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 PK S


ADEF

b) Plane UR ST and plane

XR SY

Example 3 : Plane TRQ and

plane SRQP

Angle : TRS


ABV

b) Plane PQSR and plane

PQK

A B

C D

V

J K

Q

M

R S

P

L

P Q

R

X

W

U

G H

E F

D A

B C

BA

F

E

D C

R S

U T

YX

P Q

R S

LK

P Q

R S

LK

T

R

QP




c) Plane JKLM and planePKL

d) Plane MHEL and plane NHE

e) Plane ABCD and planeBCE

Example 4: Plane DEV andDEFG

Angle : VMO

4a) Plane GCB and planeABCD

b) Plane KLMN and planeKP N

c) Plane ABE and plane

ABCD

d) Plane RUQ and plane

SRUT

e) Plane SURP and plane PTR

J K

Q

M

R S

P

L

BA

F

E

D C

G H

E F

LK

N M

D E

F G

V

O

yM AB

C D

G

O yL

BA

F

E

D C

yL

yK

R

S

T

Q

y N

yM

T

LM

N K

yE

y F

P

Q

yW

R

T

yV

S

P




Exercise 1 : Name the angle between the two planes

a) Plane SRQP and plane QUTR b) Plane JQR M and plane JKLM

c) Plane R SYX and plane UR ST d) Plane BCF and plane ABCD

e) Plane GMLF and plane GHEF f) Plane PQA and plane PQR S

C y

J K

Q

M

R S

P

L

R S

U T

YX

BA

F

E

D C

G H

E F

LK

N M

P Q

R S

X

W

V

Ay

By

T

Q

R

U

P

S




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 of

DEFG.

b) The diagram shows a cuboid with a

horizontal base JKLM . Name the angle

between the plane SRKJ and the plane of

SRLM.

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

R STU.

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 angle between the plane UVWX and the plane of

PSWV.

J K

M

R S

P

L

R S

U T

YX

BA

F

E

D C

G H

E F

LK

N M

yQ

y P




Questions According To SPM Format

1. Diagram 1 shows a pyramid with a

rectangular base PQR S. V is vertically above P.

Name the angle between the line VR and the

plane PQR S

A. PRV B. VR S

C. PVQ D. VQS

2. Diagram 2 shows a cuboid.

Name the angle between the line PM and the

plane SR MN.

A. PMK B. PMQ

C. PMR D. PMS

3. In the diagram 3, X and Y are the midpointsof PW and SV respectively.

The angle between line RX and the plane

R SVU 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. EK F

C. HEK D. JEK

K L

M N

R S

P Q

P Q

R S

V

DIAGRAM 1 DIAGRAM 2

Q P

WS

VU

R T

yX

DIAGRAM 3

yY

F G

K

H

E J

DIAGRAM 4




5. Diagram 5 shows a pyramid .PQR S is arectangle.

Name the angle between the line TQ and the

plane PQR S

A. QPT B. QST

C. R ST D. SQT


The angle between the line SU and plane

PSWT 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 and

the plane FGKJ.

A. EGF B. EK F

C. HGJ D. GK E

T U

VW

R S

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




9. Diagram 9 shows a right prism .

Name the angle between the plane ACS and the plane SADR.

A. CAS B. SCD

C. ACD D. CAD

10. Diagram 10 shows a right pyramidMABCD 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 anequilateral triangle and PV = VR = SW = WU .

The angle between the plane PRT and the planePRUS 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 plane

JKLM.

A. YBA B. AYB

C. YAB D. ABY

DIAGRAM 11

DIAGRAM 9

Q C

D R

AS

P B

Q

P

yWS

R

T

yV

J K

L

Y

X

M

DIAGRAM 12

yA

yB

DIAGRAM 10

A B

C

O




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 triangleRUQ and STP are the uniform cross-section of the prism. M and N are the mid-points of ST

and RU, respectively.

The angle between the plane PR N and the plane R STU is

A. PR S B. MRP

C. P NM D. NPM

16. Diagram 16 shows a cuboid with base

PQR S . .

Name the angle between the plane WXR andthe plane QRWV.

A. QRX B. RXQ

C. RWX D. SXU

DIAGRAM 14DIAGRAM 13

C

D

yM A

B

F

L

J

HG

R

U

S

P

T

Q

DIAGRAM 15

y N

yM R

Q

U

W

T

S

P

V

yX

DIAGRAM 16




17. Diagram 17 shows a cuboid withrectangular 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 ahorizontal plane EFGH. J is a mid-point of

GH.

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 horizontal plane QRP and vertical rectangular QR ST .

Name the angle between the plane PR S and the plane QR ST is

A. PSQ B. PST

C. PRT D. PRQ

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

JKLM .

Name the angle between line NK and base

JKLM.

A. NK M 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

L




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 and

N 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. P NQ D. MP N

2. SPM July 2004, Q16

Diagram 9 shows a cuboid with PQR S as its horizontal base.

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

A. TRW B. TQV

C. WTR D. VTQ

T

R

US yM

y N

P

P Q

R S

WT

U V

DIAGRAM 6

DIAGRAM 9




3. SPM Nov 2004, Q14

Diagram 8 shows a cuboid with horizontal base PQR S.

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 NPQR S with square base PQR S.

The angle between the line NQ and the base PQR S is

A. PQ N B. NQS

C. Q NS D. Q NR

P

R S

WT

U

Q R

S P

N

DIAGRAM 8

DIAGRAM 6




5. SPM Nov 2005, Q14

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

S P

Q R

DIAGRAM 7




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 PQR S. 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 R PS

B R P Q

C P RS

D P RQ

P

Q

R S

Diagram




9. SPM Nov 2007 Q14

Diagram 8 shows a pyramid with its rectangle base QR ST.

Vertex P is vertically above T.

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

A P QT

B PS T

C SP Q

D S TQ

10. SPM Jun 2008, Q 15.


What is the angle between the plane SVW and the plane PQR S?

A QSW

B RSW

C QS V

D PS V

P

Q

R

S

T

U

V

W

Diagram 8

P

Q

R

ST

Diagram 8




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 U S V

D S UV

P

Q S

T

U

V




A NSWER S

17.1.21a) VO , DO b) K S , SQ

2a) GH , CH b) AB , BV

3a) BC , VC b) GF , FD

17.1.3

1a) VDO b) KQS 2a) GCH b) AVB

3a) BVC b) FDG

Exercise 1a) NUM b) NJG

c) SXT d) EBL

e) K MF f) APB

Exercise 2

a) VGO b) LSK c) RYT d) EQP

e) ENF f) RVS

17.2.21a) AB b) AD

2a) K S b) KL

3a) DG b) DE 4a) AD b) VC

17.2.31a) Line SL and ML or line PK and JK are

perpendicular to LK

b) Line UV and PV or line XW and SW are perpendicular to VW

2a) Line FB and FA or line CD and ED are perpendicular to AD

b) Line XR and UR or line ST and YS are perpendicular to R S

3a) Line VB and CB is perpendicular to AB

b) Line KP andSP is perpendicular to PQ

c) Line PK and JK is perpendicular to KL

d) Line NH and MH is perpendicular to EH

e) Line EC andDC is perpendicular to BC

4a) Line GL and OL is perpendicular to BC

b) Line PE and EF is perpendicular to LM

c) Line EL and LK is perpendicular to AB

d) Line Q N andMN is perpendicular to RU

e) Line WV and TV is perpendicular to PR

17.2.4

1a) JKP or MLS

b) UVP or XWS

2a) BAF or CDE

b) URX or TSY

3a) VBC b) KPS c) PKJ

d) NHM e) ECD

4a) GLO b) PEF c) ELK

d) Q NM e) TVW

Exercise 1

a) PQU or SRT b) QJK or R MLc) XRU or YST

d) FBA e) MGH or LFE f) ACB

Exercise 2

a) VLO b) MSJ or KRL

c) YST d) EQP e) K EF f) 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. B

11. 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

chapter 17 i lines & planes in 3d enhance

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