Form 4 Chapter 11 Lines and Planes in 3-Dimensions

CHAPTER 11 LINES AND PLANES IN 3-DIMENSIONS

PRIOR KNOWLEDGE 3 Different types of dimensions

One- Dimensional A line Only has length

MATHEMATICS FORM 4

Two- Dimensional One surface length and width

Three- Dimensional  more than one surface  length, width and height

PRIOR KNOWLEDGE Trigonometric Ratios c b

a

Pythagoras’ Theorem

MATHEMATICS FORM 4

11.1 ANGLE BETWEEN LINES AND PLANES A. Identify Plane

 PLANE: is a flat surface

Plane

Not a Plane

MATHEMATICS FORM 4

11.1 ANGLE BETWEEN LINES AND PLANES  3 types of plane Horizontal plane

MATHEMATICS FORM 4

Vertical plane

Inclined plane

Horizontal plane

Vertical plane

Vertical plane

Inclined plane

Activity 1 1.

According to the prism below. Name the specific plane. E A

 Horizontal plane ABFE DHGC

F

B H G

 Inclined plane BFGC

D C MATHEMATICS FORM 4

 Vertical plane ABCD EFGH ADHE

11.1 ANGLE BETWEEN LINES AND PLANES B. Identify Lines

Lines that lie on a plane D

C

A

MATHEMATICS FORM 4

B

11.1 ANGLE BETWEEN LINES AND PLANES Lines that intersect with a plane

D

C

A

MATHEMATICS FORM 4

B

11.1 ANGLE BETWEEN LINES AND PLANES Normal to a plane Definition: Normal to a plane is a perpendicular straight line to the intersection of any lines on the plane. X Normal to a plane S

R Y

P MATHEMATICS FORM 4

Q

Activity 2 1.

Identify the normal(s) to each of the given planes. E A

Example: Normal to the plane ADHE are

F

Answer: AB, DC, EF and HG

B H

G D C MATHEMATICS FORM 4

Activity 2 (a) Normal to the plane CDHG are E

AD and HE

F

(b) Normal to the plane BCGF are A

No normal line

B H

G D C MATHEMATICS FORM 4

11.1 ANGLE BETWEEN LINES AND PLANES Orthogonal Projection Definition: Is a perpendicular projection of the object on a plane. B

Orthogonal projection of line AB on the plane PQRS

A

R

S

P

MATHEMATICS FORM 4

Q

Plane at bottom

Plane at top

Plane at right hand side

Plane at left hand side

Plane at the back

Plane in front

REMEMBER THIS…

Imagine … Screen=PLANE Object=LINE

MATHEMATICS FORM 4

Activity 3 Find the orthogonal projection of a given line on a specific plane given.

1.

D

C B

A

R

S P MATHEMATICS FORM 4

Q

Line

Plane

Orthogonal Projection

a)

AC

ADSP

AD

b)

BD

DCRS

CD

c)

AR

PQRS

PR

d)

PC

ABCD

AC

e)

QC

DCRS

RC

f)

DQ

PQRS

SQ

Angle between a line and a plane B

BC is normal to the plane PQRS

R

S A P

C

Q

Orthogonal projection of line AB on the plane PQRS is line AC

Angle between the line AB and the plane PQRS is the angle form between the line AB and the orthogonal projection on the plane.

ANSWER: ∠ B A C MATHEMATICS FORM 4

Angle between a line and a plane TECHNIQUE…

∠ __ __ __ Point NOT TOUCH on plane

MATHEMATICS FORM 4

Point TOUCH on plane

NORMAL of not touch point on plane

Identify the angle between the line AB and the plane PQRS B

R

S A P

C Q

∠ __ B __ A __ C NOT TOUCH

TOUCH

NORMAL

Activity 4 S

P

R

Q

Based on the diagram, name the angles between the following:

∠

NOT TOUCH

TOUCH

C

D

Answers A

B

(a) Line BR and plane ABCD (b) Line AS and plane ABCD (c) Line AR and plane CDSR (d) Line BS and plane PQRS

R __ B __ C ∠ __ S __ A __ D ∠ __ A __ R __ D ∠ __ B __ S __ Q ∠ __

NORMAL

Activity 5 Identify the angle of the line and the plane given.

1.

D

C B

A

R

S P MATHEMATICS FORM 4

Q

Line

Plane

Angle

a)

AC

ADSP

∠CAD

b)

BD

DCRS

∠BDC

c)

AR

PQRS

∠ARP

d)

PC

ABCD

∠PCA

e)

QC

DCRS

∠QCR

f)

DQ

PQRS

∠DQS

Example 1 S

R

Q

P P

3cm

A

3 cm

Based on the diagram, (a)Identify the angle between the line PB and the plane ABCD.

A ∠__ __ __ P B Not Touch

Touch

Normal of P

4 cm

D

C

4 cm

B

(b) Hence, calculate the angle between the line PB and the plane ABCD. tan ∠PBA =

∠PBA = tan -1 ∠PBA = 36˚52′

Example 2

S

(a) Find the angle between the line SB and the plane ABCD.

D ∠__ __ __ S B Not Touch

Touch

P

Q

Normal of S

(b) Calculate the angle between the line SB and the plane ABCD if SB = 19cm and BD= 13 cm.

R

D

C

A

B

S

cos ∠SBD = 19 cm

H

∠SBD = cos -1

∠SBD= 46˚50′

D

13 cm

A

B

Example 3 (SPM 2006 PAPER 2) T

Diagram shows a right prism. The base PQRS is a horizontal rectangle. The right angled triangle UPQ is the uniform cross section of the prism.

U 9 cm

S

R Identify and calculate the angle between the line RU and the base PQRS. [3 marks]

P 5 cm Q

U __ R __ P ∠ __

Calculate angle

U 9 cm P

Identify angle

tan ∠URP= 13 cm

R

∠URP = tan -1 ∠URP= 34˚42′

MATHEMATICS FORM 4

Example 4(SPM 2008 PAPER 2) E

Diagram shows a cuboid. M is the midpoint of the side EH and AM = 15 cm.

M

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]

H

F

G D A

C 8cm B

(a)

M __ A __ E ∠ __

(b)

sin ∠MAE=

O

∠MAE= sin -1

4cm M

E

15 cm

H A MATHEMATICS FORM 4

∠MAE= 15˚28′

Name angle

Example 5 (SPM 2007 PAPER 2) T U

S

5 cm

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]

R

V

V __ U __ S ∠ __

P

Calculate angle

Q S

13 cm

8cm

U

tan ∠VUS= ∠VUS= tan -1

V MATHEMATICS FORM 4

Identify angle

∠VUS= 31˚36′

11.2 ANGLE BETWEEN PLANES AND PLANES Identify the angle between the plane ABCD and the plane BCEF.

∠ __ A __ B __ F D __ C __ E OR ∠ __ E D D

MATHEMATICS FORM 4

E

C

F

A

C

B

A

B

F

11.1 ANGLE BETWEEN PLANES AND PLANES Identify the angle between the plane ABCD and the plane BCF.

F __ B __ A ∠ __ D

F D

A MATHEMATICS FORM 4

C

C

B

A

B

F

11.1 ANGLE BETWEEN PLANES AND PLANES Identify the angle between the plane ABC and the plane BCD.

A __ B __ F ∠ __ C F C

F A MATHEMATICS FORM 4

B

A

B

11.1 ANGLE BETWEEN PLANES AND PLANES Identify the angle between the plane ABCD and the plane BCE.

E __ F __ G ∠ __ E D D

C

G

A MATHEMATICS FORM 4

C

F

B

E G

F

A

B

SPM 2006 (PAPER 1) 1) Name the angle between the plane PQWT and the

plane SRWT. Q

R

V

W

P

S

U

T

MATHEMATICS FORM 4

P __ T __ S ∠ __ OR

Q __ W __ R ∠ __

SPM 2007 (PAPER 1) 2) Vertex P is vertically above T. Name the angle between the plane PTS and the plane PTQ. P

Q __ T __ S ∠ __ S

T

Q MATHEMATICS FORM 4

R

SPM 2008 (PAPER 1) 3) What is the angle between the plane STU and the base QSTV.

U __ T __ V ∠ __

U

P

V

Q

MATHEMATICS FORM 4

T

S

SPM 2009 (PAPER 1) 3) Given M and N is the midpoint of the line QR and PS. Name the angle between the plane VQR and the base PQRS. V

V __ M __ N ∠ __

U

S

R

N P

MATHEMATICS FORM 4

M Q

SPM 2010 (PAPER 2) Diagram in the answer space shows a right prism. The base CDHG is a horizontal rectangle. Trapezium ABCD is the uniform cross section of the prism. (a) On diagram in the answer space, mark the angle between the plane BCGF and the base CDHG . (b) Hence, calculate the angle between the plane BCGF and the base CDHG. [3 marks] B Answer : (b) E 2 cm F (a) A

Mark

B

X

H

13 cm

G

D

21 cm MATHEMATICS FORM 4

13 cm 19 cm

C

tan ∠BCD = ∠BCD= tan -1

C

∠BCD= 34˚23′

SPM 2005 (PAPER 2) Diagram shows a right prism. Right-angled triangle PQR is the uniform cross section of the prism. (a) Name the angle between the plane RTU and 12 cm T S the plane PQTU. 5 cm (b) Hence, calculate the angle between U the plane RTU and the base PQTU.

[3 marks]

Answer : (a) R

18 cm

T __ Q ∠ __ __

Q R P

(b) T 18 cm

Q MATHEMATICS FORM 4

tan ∠RTQ= ∠RTQ= tan -1 12 cm

R

∠RTQ= 33˚41′

SPM 2009 (PAPER 2) Diagram below shows a cuboid with horizontal base ABCD. J is the midpoint of the side AF. E (a) Name the angle between the plane BCJ and the base ABCD. H

F J

D

8 cm

G

A

C 6 cm

10 cm B

(b) Calculate the angle between the plane BCJ and the base ABCD. [3 marks] Answer : (a) J

B __ A ∠ __ __

(b)

tan ∠JBA=

J ∠JBA= tan -1

4 cm A MATHEMATICS FORM 4

10 cm

B

∠JBA= 21˚48′

Example (PAPER 2) Diagram shows a right pyramid. V is the vertex of the pyramid and the base PQRS is a horizontal square. M and N is the midpoint of QR and PS. The height of the pyramid is 11 cm. V Identify and calculate the angle between the plane VQR and the base PQRS. [3 marks] U

V __ M __ N ∠ __

S

R

N P

V tan ∠VMX=

M ∠VMX= tan -1

11 cm 10 cm

Q ∠VMX= 65˚33′ X 5 cm

M