chapter 9 ii lines & planes in 3d enrich

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

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CHAPTER 9 : LINES AND PLANES IN 3 DIMENSIONS

How to answer the SPM format Question

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 cm

vertically 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 = QJLJ

=13

8

LQJ =

LINE AND PLANES IN 3-DIMENSIONS 1

Q R

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

Jcm

Q R

SP

L

10 cm

12 cm

J

cm


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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 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 SPM 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 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.


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


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5) Diagram 5 shows a pyramid with the

horizontal plane, TRS. The rectangle PQRS is

vertical 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 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.


YV

WX

TS

R U

10 cm

6 cm

4 cm

Z

DIAGRAM 6

T

RS

QP

12 cm13 cm

10 cm

DIAGRAM 5

B

DIAGRAM 7 DIAGRAM 8

EF

GH

R

5 cm

24 cm

S

A

CD

FE

8 cm

6 cm

6 cm


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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 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.


DIAGRAM 9 cm

LMM

QP

RS

KNN

Y

10 cm

6 cm

12 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

QQ

MM

L

6 cm

5 cm

12 cm

DIAGRAM 12 cm


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PAST YEAR QUESTIONS

Nov 2003

1. 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 therectangular surface GFLH is incline.

Calculate the angle between the plane DLH and the base HJKL. [ 4 marks ]

July 2004, Q4

2. 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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Nov 2004, Q3

3. 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 ]

July 2005, Q2

4. 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


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Nov 2005, Q4

5. 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.

July 2006, Q4

6. 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.


U

Q

ST

P

R

12 cm

5 cm

18 cm

DIAGRAM 1

DIAGRAM 2

J

M

H

K

L

N

6 cm

12 cm

8 cm


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Nov 2006, Q2

7. 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.


P R

W

Q

12 cm

T

S

U

7 cm

14 cm

5 cm

V


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Identify and calculate the angle between the plane PQW and the base PQRS.

[3 marks]

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.


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


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Name and calculate the angle between the plane FPCR and the base ABCD.

[4 marks]

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


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ANSWERS

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

SPM PAST YEAR QUESTIONS

1 Nov 200336.87o / 36o 52

2 Jul 2004 18.43o / 18o 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 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 enrich

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