Chapter 17 I Lines & Planes in 3D ENRICH

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

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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.

P Q

RS

K

L

V

o

Line : KV

Plane : PQRS

Normal to the plane : VLOrthogonal projection of the line

onto the plane : KL

Angle between KV and the plane

PQRS is VKL

LM and MN are perpendicular toDC.

The angle between the planeABCD and the plane CDEF is

KML ( or ADE orBCF )

M

L

NAB

CD

E F


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17.2.1 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 PKS

2. a) Plane ABCD and plane

ADEF

b) Plane URST and plane

XRSY

Example 3 : Plane TRQ and

plane SRQP

Angle : TRS

3. a) Plane ABCD and plane

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

yM A B

CD

G

O yL

BA

F

E

D C

yL

yK

R

S

T

Q

yN

yM

T

LM

NK

yE

y F

P

Q

yW

R

T

yV

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

y

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

yQ

y P


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Questions Based On Examination 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 the

plane PQRS

A. PRV B. VRS

C. PVQ D. VQS

2. Diagram 2 shows a cuboid.

Name the angle between the line PM and the

plane SRMN.

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

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

yX

DIAGRAM 3

yY

F G

K

H

E J

DIAGRAM 4


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5. Diagram 5 shows a pyramid .PQRS is arectangle.

Name the angle between the line TQ and the

plane PQRS

A. QPT B. QST

C. RST 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. 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 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

DR

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


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PAST YEAR SPM QUESTIONS

Nov 2003, Q13

1. 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. PNQ D. MPN

July 2004, Q16

2. 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

T

R

US yM

yN

P

P Q

RS

WT

U V

DIAGRAM 6

DIAGRAM 9


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Nov 2004, Q14

3. 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

July 2005, Q14

4. 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

P

RS

WT

U

Q R

SP

N

DIAGRAM 8

DIAGRAM 6


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Nov 2005, Q14

5. 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

July 2006, Q14

6. 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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Nov 2006, Q14

7. Diagram 7 shows a cuboid with a horizontal base TUVW.

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 Pis 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.

Diagram 8 shows a cuboid with a horizontal base TUVW.

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


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Line And Planes In 3 Dimensions 18

ANSWERS

17.2.1

1a) JKP or MLS

b)

UVP or

XWS

2a) BAF or CDEb) URX or TSY

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

4a) GLO b) PEF c) ELKd) QNM e) TVW

Exercise 1

a)

PQU or

SRTb) 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) KEF

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. 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. B5. C 6. A 7. B 8. A

9. D 10. B 11. B

Documents

Chapter 17 I Lines & Planes in 3D ENRICH