Chapter 11 - Line and Planes in 3-Dimension

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Lesson 104: Horizontal Planes, Vertical Planes and Inclined Planes.

Plane Type of planes

EDIJ

ABGF

BGHC

ABCDE

CHID


ABCD

AEB

BCFE

B

G

H

I

J

E

AD

F

C

Based on the figure given above, identify the type of planes for:

Question 1

A

B

E

F C

D

Question 2


Mathematics Form 4

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AFB

CFD

ABCD


ABFE

EFGH

ABCD

Question 3

A B

C D

F

E

Question 4

A B

F E

C

G H

D



Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy Lesson 104: Horizontal Planes, Vertical Planes and Inclined Planes.


EDIJ Inclined plane

ABGF Horizontal plane

BGHC Vertical plane

ABCDE Vertical plane

CHID Horizontal plane


ABCD Vertical plane

AEB Horizontal plane

B

G

H

I

J

E

AD

F

C


Question 1

A

B

E

F C

D

Question 2


Mathematics Form 4

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BCFE Vertical plane


AFB Inclined plane

CFD Inclined plane

ABCD Horizontal plane


ABFE Vertical plane

EFGH Horizontal plane

ABCD Horizontal plane

Question 3

A B

C D

F

E

Question 4

A B

F E

C

G H

D



Mathematics Form 4

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ACTIVITY SHEET Pupils’ Copy Lesson 105: Lines on a Plane, Lines Intersect with a Plane and Normals.

A D

C B

F G

H E

1. Name all the lines that lie on the plane CDHG. Answer: 2. Name all the lines that intersect with the plane BCGF. Answer : 3. Name all the normals to the plane ABFE Answer :

Question 1

Q

R P

S U

T

V

W

1. Name all the lines that lie on the plane RQTU. Answer: 2. Name all the lines that intersect with the plane PQTS. Answer : 3. Name all the normals to the plane STU Answer :

Question 2

1. Name all the lines that lie on the plane ABHG. Answer: 2. Name all the lines that intersect with the plane FAGL. Answer : 3. Name all the normals to the plane ABCDEF Answer :

Question 3

1. Name all the lines that lie on the plane ABCD. Answer: 2. Name all the lines that intersect with the plane LKOP. Answer : 3. Name all the normals to the plane EFGH Answer :

Question 4

A B

C

D E

F G H

I

J K

L

A

B

C

D H

E

G

F I

J

K L

M

P O

N

Mathematics Form 4

2

ACTIVITY SHEET Teacher’s Copy Lesson 105: Lines on a Plane, Lines Intersect with a Plane and Normals.

A D

C B

F G

H E

1. Name all the lines that lie on the plane CDHG. Answer: 2. Name all the lines that intersect with the plane BCGF. Answer : 3. Name all the normals to the plane ABFE Answer :

Line: CD, DH, HG, GC, DG

Line: BA, CD, GH, FE, GD

Line: AD, BC, FG, EH

Question 1

Q

R P

S U

T

V

W

1. Name all the lines that lie on the plane RQTU. Answer: 2. Name all the lines that intersect with the plane PQTS. Answer : 3. Name all the normals to the plane STU Answer :

Line: RQ, QT, TU, UR, WV

Line: QR, VW, TU, PR, SU

Line: SP, UR, TQ

Question 2

1. Name all the lines that lie on the plane ABHG. Answer: 2. Name all the lines that intersect with the plane FAGL. Answer : 3. Name all the normals to the plane ABCDEF Answer :

Line: AB, BH, HG, GA

Line: AB, GH, LK, FE

Line: AG, BH, CI, DJ, EK, FL

Question 3

1. Name all the lines that lie on the plane ABCD. Answer: 2. Name all the lines that intersect with the plane LKOP. Answer : 3. Name all the normals to the plane EFGH Answer :

Line: AB, BC, CD, DA, IJ, JK, KL, LI

Line: LI, KJ, PM, ON

Line: EA, FB, GC, HD

Question 4

A B

C

D E

F G H

I

J K

L

A

B

C

D H

E

G

F I

J

K L

M

P O

N

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 106: Orthogonal projection of a line on a plane.

1. The figure shows a triangular prism PQRSTU. Draw and name the orthogonal projection of each of the following lines on the plane PQRS.

T

P Q

R S

U

M

N

a) TQ

b) QU

Mathematics Form 4

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2. The figure shows a cube ABCDEFGH . Based on the diagram given, complete the table below.

Line

Plane

Orthogonal Projection

Angle between line

and plane

1

AG

ABCD

2

HB

ABEF

3

CH

EFGH

4

EG

DCHG

5

HB

BCFG

A B

C D

E F

GH

Mathematics Form 4

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3. The figure shows a pyramid ABCDEFG. Based from diagram given, complete the table below.

No

Line

Plane

Angle between line and plane

1

ABCD

EFG or GFE

2

ABCD

EAC or CAE

3

ABCD

EBD or DBE

4

ABCD

EGF or FGE

5

ABCD

EDB or BDE

B A

C D

E

F

G

Mathematics Form 4

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ACTIVITY SHEET Teacher’s Copy LESSON 106: Orthogonal projection of a line on a plane.

1. The figure shows a triangular prism PQRSTU. Draw and name the orthogonal projection of each of the following lines on the plane PQRS.

T

P Q

R S

U

M

N

a) TQ

b) QU

T

M Q

U

MQ or QM

NQ or QN

N Q

Mathematics Form 4

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2. The figure shows a cube ABCDEFGH . Based on the diagram given, complete the table below.

Line

Plane

Orthogonal Projection

Angle between line

and plane

1

AG

ABCD

AC

GAC

2

HB

ABEF

BF

HBF

3

CH

EFGH

HG

CHG

4

EG

DCHG

HG

EGH

5

HB

BCFG

FB

HBF

A B

C D

E F

GH

Mathematics Form 4

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No

Line

Plane

Angle between line and plane

1

EF

ABCD

EFG or GFE

2

EA

ABCD

EAC or CAE

3

EB

ABCD

EBD or DBE

4

EG

ABCD

EGF or FGE

5

ED

ABCD

EDB or BDE

B A

C D

E

F

G

Mathematics Form 4

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ACTIVITY SHEET

Pupil’s Copy

LESSON 107: Problems Solving on Angle between a Line and a Plane.

The diagram shows a cuboid. KG = 2HK.

DL = HK. By sketching an appropriate right-angled triangle, find the following:

(a) the length of FD.

(b) the length of KA.

(c) the angle between FD and BCGF.

(d) the angle between KA and EFGH.

A B

DE F

GH

C

9 cm

4 cm

6 cm

K

L

Mathematics Form 4

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ACTIVITY SHEET

Teacher’s Copy

LESSON 107: Problems Solving on Angle between a Line and a Plane.

The diagram shows a cuboid. KG = 2HK. DL = HK. By sketching an appropriate right-angled triangle, find the following:

(a) the length of FD.

22 BDBFFD +=

)(4 222 ADAB ++=

22 6916 ++=

133= = 11.53 cm

(b) the length of KA.

22 ALKLKA +=

)(4 222 DLAD ++=

22 3616 ++= = 7.81 cm

(c) the angle between FD and BCGF.

The orthogonal projection of FD on BCGF is FC. So, the angle between FD and BCGF is ∠CFD.

DFCD

CFD =∠sin

53.11

9=

= 0.7806 ∠CFD = 51.32° or 51° 19'

(d) the angle between KA and EFGH.

The orthogonal projection of KA on EFGH is KE. So, the angle between KA and EFGH is ∠EKA.

AKAE

EKA =∠sin

81.74=

= 0.5122 ∠EKA = 30.81° or 30° 49'

A B

DE F

GH

C

9 cm

4 cm

6 cm

K

F

D B

K

A L

L

F

D C

11.53 cm

9 cm

A

E K

7.81 cm 4 cm

Mathematics Form 4

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ACTIVITY SHEET Pupil’s Copy Lesson 108: Line of Intersection between Two Planes.

Plane A Plane B Line of intersection

EDIJ DCHI

BFIC ABGF

AGH BCHG

ABCDE DEJI

DCG BCHG

Plane A Plane B Line of intersection Pair of lines perpendicular to line of intersection

ABCD BCFE

AFE BCFE

BED ABE

B

G

H

I

J

E

AD

F

C

Based on figure given above, state the line of intersection between the following planes:

Question 1

A

B

E

F C

D

Question 2

Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection.

Mathematics Form 4

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AFC ABCD

BEF BCE

BDF ABCD


AEGC ABFE

BHD ADHE

EFGH ABFE

Question 3

A B

C D

F

E


Question 4

Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection

A B

F E

C

G H

D

Mathematics Form 4

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ACTIVITY SHEET Teacher’s Copy Lesson 108: Line of Intersection between Two Planes.

Plane A Plane B Line of intersection

EDIJ DCHI Line DI

BFIC ABGF Line BF

AGH BCHG Line GH

ABCDE DEJI Line ED

DCG BCHG Line CG


ABCD BCFE Line BC Line AB & Line BE or Line CD & Line CF

AFE BCFE Line EF Line EA & Line EB

BED ABE Line BE Line EA & Line ED

B

G

H

I

J

E

AD

F

C

Based on figure given above, state the line of intersection between the following planes:

Question 1

A

B

E

F C

D

Question 2


Mathematics Form 4

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AFC ABCD Line AC Line EF & Line BD

BEF BCE Line BE Line EC & Line EF

BDF ABCD Line BD Line EF & Line AC


AEGC ABFE Line AE Line EF & Line EH or Line AB & Line AD

BHD ADHE Line DH Line AD & Line BD

EFGH ABFE Line EF Line EA & Eh or Line FB & Line FG

Question 3

A B

C D

F

E


Question 4

Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection

A B

F E

C

G H

D

Mathematics Form 4

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ACTIVITY SHEET Pupil’s Copy LESSON 109: Angle between Two Planes.

1. Based on the diagram given, identify the angle between the two planes

given by completing the table below.

No

Plane

Plane

Angle between two p lanes

1

UON

MNPQ

2

ROQ

MNPQ

3

NORQ

MORP

4

RSU

MORP

5

TQU

NORQ

M N

O

P Q

R

S

T U

Mathematics Form 4

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2. The figure shows a cuboid ABCDPQRS. Complete the table given in order to identify the the angle between two planes.

No

Plane

Plane

Angle between two planes

1

CDS

ABCD

2

PQB

PQRS

3

ABC

ABPQ

4

SRP

DCSR

5

ABPQ

CBRQ

Q P

R S

B A

C D

Mathematics Form 4

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No

Plane

Plane


1

ABCD EFG or GFE

2

ABCD EGF or FGE

3

ABCD EHI or IHE

4

ABCD EIH or HIE

B A

C D

E

F

G

H I

Mathematics Form 4

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ACTIVITY SHEET Teacher’s Copy LESSON 109: Angle between Two Planes.

1. Based on the diagram given, identify the angle between the two planes

given by completing the table below.

No

Plane

Plane


1

UON

MNPQ

ONM or MNO

2

ROQ

MNPQ

RQP or PQR

3

NORQ

MORP

UST or TSU, NOM or MON,

QRP or PRQ

4

RSU

MORP

UST or TSU

5

TQU

NORQ

SUT or TUS

M N

O

P Q

R

S

T U

Mathematics Form 4

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2. The figure shows a cuboid ABCDPQRS. Complete the table given in

order to identify the the angle between two planes.

No

Plane

Plane


1

CDS

ABCD

SDA or ADS

2

PQB

PQRS

RQB or BQR

3

ABC

ABPQ

CBQ or QBC

4

SRP

DCSR

DSP or PSD

5

ABPQ

CBRQ

RQP or PQR, CBA or ABC

Q P

R S

B A

C D

Mathematics Form 4

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No

plane

Plane


1

EAB

ABCD

EFG or GFE

2

ECD

ABCD

EGF or FGE

3

EAD

ABCD

EHI or IHE

4

ECB

ABCD

EIH or HIE

B A

C D

E

F

G

H I

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Lesson 110: Problems Solving on Lines and Planes in 3-Dimensional Shapes.

The figure shows two wooden blocks attached together. Based on the figure, solve the questions given:

Question 1

Question 2

The figure shows a prism. M and N are the midpoints of AB and DE respectively.

A B

C D K

J

M

N

L

G I

F E

H 15cm

40cm

50cm

55cm

20cm

1. What is the length of the diagonal DI?

2. Determine the angle between planes ILNK and CILG.

3. Determine the angle between horizontal plane DCGH and the diagonal DF.

4. Determine the angle between planes LGKJ and CGFB.

A B

B

M

E N

F

D

16cm

16cm

8cm

1. What is the length of the diagonal AE?

2. Identify the angle between planes ACFD and ABED.

3. Determine the angle between the diagonal BF and the plane ABED.

Mathematics Form 4

2

ACTIVITY SHEET Teacher’s Copy Lesson 110: Problems Solving on Lines and Planes in 3-Dimensional Shapes.

The figure shows two wooden blocks attached together. Based on the figure, solve the questions given:

Question 1

Question 2

The figure shows a prism. M and N are the midpoints of AB and DE respectively.

A B

C D K

J

M

N

L

G I

F E

H 15cm

40cm

50cm

55cm

20cm

1. What is the length of the diagonal DI?

2. Determine the angle between planes ILNK and CILG.

3. Determine the angle between horizontal plane DCGH and the diagonal DF.

4. Determine the angle between planes LGKJ and CGFB.

25cm

69.44°

42.0°

36.02°

A B

B

M

E N

F

D

16cm

16cm

8cm

1. What is the length of the diagonal AE?

2. Identify the angle between planes ACFD and ABED.

3. Determine the angle between the diagonal BF and the plane ABED.

18.87cm

58.0°

25.52°

Chapter 11 - Line and Planes in 3-Dimension

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Transcript of Chapter 11 - Line and Planes in 3-Dimension