Learning Area



Learning Area 11: LINES AND PLANES IN 3 DIMENSIONS Worksheet 1.1

Learning Objective 11.1: Understand and use the concept of angle between lines and planes to

solve problems.

Learning Outcome (ii) Identify horizontal planes, vertical planes and inclined planes.

Activity 1: Determine base, horizontal planes, vertical planes and inclined planes.

|Solid |Base |Horizontal plane |Vertical plane |Inclined plane |

|1 | | | | |

| | | | | |

| |PQRS |PQRS |PSUT, |TURQ |

| | | |PQT, | |

| | | |USR | |

|2 | | | | |

| | | | |VBC, |

| | | | |VDA, |

| |ABCD |ABCD |- |VAB’ |

| | | | |VDC |

| | | | | |

|3 | | | | |

| |PQRS |PQRS, |UPQV, | |

| | |TUVW |TSRW, |- |

| | | |VQRW, | |

| | | |UPST | |

| | | | | |

|4 | | | | |

| | | |UQRV, | |

| |PQRS |PQRS |TPSW, |TUVW |

| | | |TPQU, | |

| | | |WSRV | |

| | | | | |

| | | | | |

|5 | | | | |

| | | | | |

| | | | | |

| |ABCD |ABCD |VDC |VAB, |

| | | |VAD |VBC |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

Learning Area 11: LINES AND PLANES IN 3 DIMENSIONS Worksheet 2.1

Learning Objective 11.1: Understand and use the concept of angle between lines and

planes to solve problems.

Learning Outcome (v) Identify normal to a given plane.

(vi) Determine the orthogonal projection of a line on a plane.

Activity 1 : Determine right-angled triangles.

|Triangle |Right-angled triangle |Not right-angled triangle |

|1 | | |

| | | |

| | |∕ |

| | | |

| | | |

|2 | | |

| | | |

| |∕ | |

| | | |

|3 | | |

| | | |

| | | |

| |∕ | |

| | | |

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|4 | | |

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| |∕ | |

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|5 | | |

| | | |

| | | |

| |∕ | |

| | | |

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|6 | | |

| | | |

| | | |

| | |∕ |

| | | |

Activity 2 : Draw right-angled triangles.

| |Draw horizontal line from A |Draw vertical line from B |Right- angled triangle |

|1 | | | |

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|2 | | | |

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|3 | | | |

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|4 | | | |

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Activity 3 : Identify and name the normal and orthogonal projection of a line PV on the plane PQRS in each of the following objects

|Solid |Normal |Orthogonal Projection |

| | | |

| | | |

|1. | | |

| |VS |SP |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

|2. | | |

| |VR |RP |

| | | |

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| | | |

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| | | |

|3. | | |

| | | |

| | | |

| |VQ |PQ |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

|4. | | |

| |VS |SP |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

|Solid |Normal |Orthogonal Projection |

| | | |

|5. | | |

| | | |

| |VR |PR |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

|6. | | |

| | | |

| |VQ |PQ |

| | | |

| | | |

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| | | |

|7. | | |

| | | |

| |VO |PO |

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|8. | | |

| | | |

| | | |

| |VR |PR |

| | | |

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| | | |

Learning Area 11 :LINES AND PLANES IN 3 DIMENSIONS Worksheet 3.1

Learning Objective 11.1: Understand and use the concept of angle between lines and planes to solve

problems.

Learning Outcome (viii) Determine the angle between a line and a plane.

Activity 1 : Determine the angle between a line and a plane.

|Solid |Normal |Orthogonal |Right- angled triangle |Name the angle |

| | |Projection | | |

| | | | | |

| | | | | |

|1. | | | | |

| | | | | |

| |VQ |PQ | |VPQ |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|Line PV and Plane PQRS | | | | |

|2. | | | | |

| | | | | |

| | | | | |

| |WR |QR | | |

| | | | |WQR |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|Line QW and Plane PQRS | | | | |

|3. | | | | |

| | | | | |

| | | | | |

| |UP |PR | | |

| | | | |URP |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|Line UR and Plane PQRS | | | | |

|Solid |Normal |Orthogonal |Right- angled triangle |Name the angle |

| | |Projection | | |

|4. | | | | |

| | | | | |

| | | | | |

| |TS |SP | | |

| | | | |TPS |

| | | | | |

| | | | | |

|Line PT and Plane PQRS | | | | |

|5. | | | | |

| | | | | |

| | | | | |

| |WR |PR | | |

| | | | |WPR |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|Line PW and Plane PQRS | | | | |

|6. | | | | |

| | | | | |

| | | | | |

| |UP |PQ | | |

| | | | |UQP |

| | | | | |

| | | | | |

| | | | | |

|Line UQ and Plane PQRS | | | | |

|7. | | | | |

| | | | | |

| | | | | |

| |TJ |JK | | |

| | | | |TKJ |

| | | | | |

| | | | | |

| | | | | |

|Line TK and Plane JKLM | | | | |

|8. | | | | |

| | | | | |

| | | | | |

| |RL |KL | | |

| | | | |RKL |

| | | | | |

| | | | | |

|Line KR and Plane JKLM | | | | |

|Solid |Normal |Orthogonal |Right- angled triangle |Name the angle |

| | |Projection | | |

|9. | | | | |

| | | | | |

| | | | | |

| |PJ |JK | | |

| | | | |PKJ |

| | | | | |

| | | | | |

|Line PK and Plane JKLM | | | | |

|10. | | | | |

| | | | | |

| | | | | |

| |VM |MK | | |

| | | | |VKM |

| | | | | |

| | | | | |

| | | | | |

|Line KV and Plane JKLM | | | | |

| | | | | |

|11. | | | | |

| | | | | |

| |VM |ML | | |

| | | | |VLM |

| | | | | |

| | | | | |

| | | | | |

|Line VL and Plane JKLM | | | | |

|12. | | | | |

| | | | | |

| | | | | |

| |VM |MJ | | |

| | | | |VJM |

| | | | | |

| | | | | |

| | | | | |

|Line JV and Plane JKLM | | | | |

|13. | | | | |

| | | | | |

| | | | | |

| |TP |PR | | |

| | | | |TRP |

| | | | | |

| | | | | |

|Line TR and Plane PQRS | | | | |

| | | | | |

| | | | | |

Learning Area 11: LINES AND PLANES IN 3-DIMENSIONS Worksheet 4.1

Learning Objective 11.2: Understand and use the concept of angle between two planes to solve

problems.

Learning Outcome (i) Identify the line of intersection between the two planes.

Activity 1: Grouping in pairs to identify the line of intersection.

1. Identify the line of intersection between the two given planes:

| |

|Plane ABCD and plane BCHG |Plane ABCD and plane ADEF |

| | |

| | |

| | |

| | |

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| | |

| | |

|Step 1: Identify two letters in common on the |Step 1: Identify two letters in common on the |

|plane ABCD and plane BCHG: |plane ABCD and plane ADEF: |

|= B and C |= ……… and ……… |

| | |

|Step 2: The line of intersection between the |Step 2: The line of intersection between the |

|two planes: |two planes: |

|= BC |= ……….. |

|Plane CDEH and plane EFGH |Plane EFGH and plane ABGF |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|Step 1: Identify two letters in common on the |Step 1: Identify two letters in common on the |

|plane CDEH and plane EFGH: |plane EFGH and plane ABGF: |

|= …E and ……H… |= ……… and ……… |

| | |

|Step 2: The line of intersection between the |Step 2: The line of intersection between the |

|two planes: |two planes: |

|= ……EH….. |= ……….. |

2. Identify the line of intersection between the two given planes:

|i. | |Plane ABCD and plane BCFE |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

| | | |

| | |Plane ABCD and plane ADFE |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

| | | |

| | |Plane ADFE and plane ABE |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

|ii. | |Plane ABC and plane ACD |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

| | | |

| | |Plane BCD and plane ABD |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

| | | |

| | |Plane ADC and plane ABD |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

|iii. | |Plane VAB and plane ABCD |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

| | | |

| | |Plane VBC and plane ABCD |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

| | | |

| | |Plane VBD and plane ABCD |

| | |i) Common vertices: ………and ……… |

| | | |

| | |ii) Line of intersection: ……………….. |

Learning Area 11: LINES AND PLANES IN 3-DIMENSIONS Worksheet 6.1

Learning Objective 11.2: Understand and use the concept of angle between two planes to solve

problems.

Learning Outcome (ii) Draw a line on each plane which is perpendicular to the line of intersection

of the two planes at a point on the line of intersection.

Activity 1: Thinking Smart

Draw two straight lines, one on each planes stated, which are perpendicular to the line of intersection of the two planes at the point given. (You may work in pairs if necessary)

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|Example: |(a) The plane PKNS with the plane KLMN |

|The plane NSRM with the plane KLMN at point M. |at point K. |

| | |

| | |

| | |

| | |

|State: |State: |

|Line of intersection: …………….. |i) Line of intersection: …………….. |

| | |

|Line which is perpendicular to the |ii) Line which is perpendicular to the |

|line of intersection at point M: |line of intersection at point K: |

|………….… and ……………… |………….… and ……………… |

|Draw the lines as stated above: |Draw the lines as stated above: |

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| | |

|The plane PQRS with the plane LMRQ |The plane KNRQ with the plane QLMR |

|at point R. |at point Q. |

| | |

|State: |State: |

| | |

|i) Line of intersection: …………….. |i) Line of intersection: …………….. |

| | |

|ii) Line which is perpendicular to the |ii) Line which is perpendicular to the |

|line of intersection at point R: |line of intersection at point Q: |

| | |

|………….… and ……………… |………….… and ……………… |

| | |

|Draw the lines as stated above: |Draw the lines as stated above: |

| | |

| | |

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| | |

| | |

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| | |

| | |

Activity 2: Do you know how to draw?

Draw the pair of lines perpendicular to the line of intersection for each of the pairs of planes given below:

|1. | | |

| | |(a) ADHE and EFGH |

|2. | | |

| | |(b) CDFE and ADF |

|3. | | |

| | |(a) PXZ and ABCD. |

| | | |

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Learning Area 11: LINES AND PLANES IN 3-DIMENSIONS Worksheet 6.1

Learning Objective 11.2: Understand and use the concept of angle between two planes to solve problems.

Learning Outcome (iii) Determine the angle between two planes on a model and a given diagram.

Activity 1:

Determine the angle between two given planes in each of the following:

|1. | | |

| | |(a) EFGH and BCGF: |

| | | |

| | |i) Line of intersection: ...………………… |

| | | |

| | |ii) Angle between the planes: ………………. |

| | | |

| | |(b) ABGH and ABCD: |

| | | |

| | |i) Line of intersection: ...………………… |

| | | |

| | |ii) Angle between the planes: ………………. |

|2. | | |

| | |(a) ABEF and ABCD: |

| | | |

| | |i) Line of intersection: ...………………… |

| | | |

| | |ii) Angle between the planes: ………………. |

| | | |

| | |(b) AEC and BCE: |

| | | |

| | |i) Line of intersection: ...………………… |

| | | |

| | |ii) Angle between the planes: ………………. |

|3. | | |

| | |(a) VBC and ABCD: |

| | | |

| | |i) Line of intersection: ...………………… |

| | | |

| | |ii) Angle between the planes: ………………. |

| | | |

| | |(b) VBO and VAO: |

| | | |

| | |i) Line of intersection: ...………………… |

| | | |

| | |ii) Angle between the planes: ………………. |

Learning Area 11: LINES AND PLANES IN 3-DIMENSIONS Worksheet 7.1

Learning Objective 11.2: Understand and use the concept of angle between two planes to solve

problems.

Learning Outcome (iv) Solve problems involving lines and planes in 3-dimensional shapes.

STEPS TO FOLLOW:

Step 1: Sketch the shape by referring to the information given.

Step 2: (a) Identify the line of intersection between the two planes.

(b) Draw the pair of perpendicular lines to the line of intersection.

(c) Determine the angle between the two given planes.

Step 3: Sketch the right angled triangle and label all the values given.

Step 4: Calculate the “angle” by using the Pythagoras’ theorem and trigonometry ratio

Example:

The figure shows a cuboid.

Task:

Calculate the angle between

the planes BCHE and ABCD.

Solution:

Step 1:

Step 3: Step 4:

E

A B

Guided Exercise:

Guided Solution:

a) Using Pythagoras’ Theorem,

DE = 12 cm, EG = 15 cm

EH = [pic]

= ……………..

b) Step 1: Step 2:

Identify the angle between the two planes (EHI and DGHI)

……………

Step 3: Step 4:

Sketch the right angled triangle Using Trigonometry Ratio to

in Step 2. calculate the answer.

Notes:

1. Letters labelled in the diagram given are called vertices.

The line of intersection of two planes is the line joining the common vertices in the two

planes.

2. Two planes intersect at a straight line. The straight line is called the line of intersection

of the two planes.

3.

4.

-----------------------

V

C

B

A

A

B

A

B

A

A

A

A

A

B

B

B

B

B

[pic]

P

V

Q

D

A

A

A

R

Q

R

P

U

W

[pic][pic]

[pic][pic][pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

B

B

B

A

A

A

B

B

B

B

B

B

A

A

A

E

H

F

D G

C

A

B

H

B

C

D

G

A

E

D

F

C

A

B

D

A

AD

E

F

H

G

G

D

C

E

B

H

A

F

G

G

F

FG

F

D

B

E

A

C

C

B

BC

D

A

AD

E

A

AE

A

B C

D

C

A

AC

D

B

BD

D

A

AD

V

D

A

C

B

A

B

AB

C

B

BC

D

B

BD

M

R

L

K

Q

N

S

P

R

S

N

M

K

L

P

N

M

L

S

K

NM

RM

LM

KN

KL

PK

S

R

N

M

K

L

P

N

K

M

L

S

P

Q

R

S

L

M

Q[pic]

N

M

R

K

L

QR

RM

SR

QR

QL

KQ

P

Q

R

S

M

L

R

Q[pic]

N

M

L

K

F

H

B

A C

D

G

E

A

F

H

G[pic]

D

E

E

F C

B

D

A

E

F C

D

A

B

P

A

C

X

Z

C

W

Y

O

D

P

A

B

X

Z

D

C

W

Y

O

H G

E F

D C

A B

FG

or

[pic]BFE

[pic]CGH

AB

or

[pic]CBG

[pic]DAH

E

F C

B

D

A

AB

or

[pic]DAF

[pic]CBE

CE

[pic]ACB

V

D

C

P O

N

A B

BC

[pic]VNP

VO

[pic]AOB

H

E D G

A C

B

10 cm

5 cm

5 cm

H

E D

A C

B

Step 2:

The angle between the planes

BCHE and ABCD is

[pic]ABE or [pic]DCH.

tan ¸ = [pic]

tan [pic]

tan [pic]ABE = [pic]

= 0.5

[pic][pic]ABE = 26.57o or 26o34’

10 cm

5 cm

Using scientific calculator

F

The figure shows a right prism with a uniform trapezium DEJI as its cross section.

Calculate:

a) the length of EH,

b) the angle between the planes

EHI and DGHI.

6 cm

H K

10 cm

I J

G F

9 cm

D E

8 cm

15 cm

H

E

8 cm

G

17 cm

H

I

G

D E

I

D

E

[pic]

I

D

E

[pic]

tan [pic]DIE = [pic]

= 1.5

[pic][pic]DIE = 56.32o or 56o19’

O

Plane 2

Plane 1

line of intersection

12 cm

D M C

Line of intersection is XOY.

Lines perpendicular to XOY are XD and XA, OM and ON or YC and YB.

X O Y

A N B

D M C

A N B

X O Y

The angle between the lines perpendicular to the line of intersection of two planes is the angle between the two planes.

Lines perpendicular to the line of intersection

Line of

intersection

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