Copy Lines & Planes

7/29/2019 Copy Lines & Planes

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Program Waja Negeri Perak Tahun 2009

FORM 4 MATHEMATICS Module 5 : Page 1

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,

PQT,

USR

TURQ

2

ABCD ABCD -

VBC,

VDA,

VAB

VDC

3

PQRS PQRS,

TUVW

UPQV,

TSRW,

VQRW,

UPST

-

4

PQRS PQRS

UQRV,

TPSW,

TPQU,

WSRV

TUVW

5

ABCD ABCD VDC

VAD

VAB,

VBC

A B

C

V

D


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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

4

5

6


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Activity 2 :Draw right-angled triangles.

Draw horizontal line

from A

Draw vertical line from

B

Right- angled triangle

1

2

3

4

A

B

A

A

A

A

AA

B

B

BBB

B

AA A

BBB

AAA

B B B

B B B

A A A


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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

3.

VQ PQ

4.

VS SP


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Solid Normal Orthogonal Projection

5.

VR PR

6.

VQ PQ

7.

VO PO

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

Projection

Right- angled triangle Name the

angle

1.

Line PV and Plane PQRS

VQ PQ VPQ

2.

Line QW and Plane PQRS

WR QR

WQR

3.

Line UR and Plane PQRS

UP PR

URP

P

V

Q

R

Q R

P

U

W


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Projection

Right- angled triangle Name the angle

4.

Line PT and Plane PQRS

TS SP

TPS

5.

Line PW and Plane PQRS

WR PR

WPR

6.

Line UQ and Plane PQRS

UP PQ

UQP

7.

Line TK and Plane JKLM

TJ JK

TKJ

8.

Line KR and Plane JKLM

RL KL

RKL

T

SP

R

QP W

RP

T

SP

R

LK

P

U

Q

J

T

K


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Projection

Right- angled triangle Name the angle

9.

Line PK and Plane JKLM

PJ JK

PKJ

10.

Line KV and Plane JKLM

VM MK VKM

11.

Line VL and Plane JKLM

VM ML

VLM

12.

Line JV and Plane JKLM

VM MJ

VJM

13.

Line TR and Plane PQRS

TP PR

TRP

J

P

K

M

V

K

M

V

L

V

MJ

P

T

R


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Learning Area11: 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:

(a) PlaneABCD and planeBCHG

Step 1: Identify two letters in common on theplaneABCD and planeBCHG:

=B and C

Step 2: The line of intersection between the

two planes:

=BC

(b) PlaneABCD and planeADEF

Step 1: Identify two letters in common on theplaneABCD and planeADEF:

= and


two planes:

= ..

(c) Plane CDEHand planeEFGH

Step 1: Identify two letters in common on the

plane CDEHand planeEFGH:

= Eand H


two planes:

= EH..

(d) PlaneEFGHand planeABGF

Step 1: Identify two letters in common on the

planeEFGHand planeABGF:

= and


two planes:

= ..

B

A

E

F

F

E

C

D

A

H

B

C

DG

C

D G

E

HG

B

H

A

FG

E

H

F

D G

C

AB

A D

AD

F G

FG


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

i.

(a) PlaneABCD and planeBCFE

i) Common vertices: and

ii) Line of intersection: ..

(b) PlaneABCD and planeADFE



(c) PlaneADFEand planeABE



ii.

(a) PlaneABCand planeACD



(b) PlaneBCD and planeABD



(c) PlaneADCand planeABD



iii.

(a) Plane VAB and plane ABCD



(b) Plane VBCand planeABCD



(c) Plane VBD and planeABCD



E

A

A

B C

D

V

D

AB

C

F

D

B

B C

BC

A D

AD

A E

AE

A C

AC

B D

BD

A D

AD

A B

AB

B C

BC

B D

BD

C

O


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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)

Example:

The planeNSRMwith the planeKLMNat

pointM.

(a) The planePKNS with the planeKLMN

at pointK.

State:

i) Line of intersection: ..

ii) Line which is perpendicular to the

line of intersection at point M:. and

State:



line of intersection at pointK:. and

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

S R

N M

K L

NM

LM RM

K

S

P

NM

L

M

R

LK

Q

N

S

P

S R

N M

K L

P

N

K

M

L

S

KN

PK KL


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(b) The planePQRS with the planeLMRQ

at pointR.

(c) The planeKNRQ with the plane QLMR

at point Q.

State:



line of intersection at pointR:

. and

State:



line of intersection at point Q:

. and

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

P Q

RS

L

M

L

P Q

RS

M

L

Q

N M

R

K

K

QR

SR RM

QR

KQ QL

R

Q

N M

L


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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)ADHEand EFGH

2

.

(b) CDFEandADF

3.

(a)PXZand ABCD.

D

P

B

A C

D

G

E

F

H

E

F C

B

D

A

E

F C

D

A

B

P

A

C

XZ

C

W

Y

O

A

F

H

G

D

E

A

B

XZ

D

C

W

Y

O


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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)EFGHandBCGF:

i) Line of intersection: ...

ii) Angle between the planes: .

(b)ABGHand ABCD:



2

.

(a)ABEFand ABCD:



(b) AECand BCE:



3

.

(a) VBCand ABCD:



(b) VBO and VAO:



H G

E F

D C

A B

E

F CB

D

A

V

D

C

P O

N

A B

FG

BFE

CGH

or

AB

AB

CE

BC

VO

CBG

DAH

DAFCBE

ACB

VNP

AOB

or

or


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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 betweenthe planesBCHEandABCD.

Solution:

Step 1:

Step 3: Step 4:

E

A B10 cm

5 cm

tan ABE=10

5

= 0.5

ABE = 26.57o or 26o34

Using scientific

calculator

Step 2:

The angle between the planes

BCHEandABCD is

ABE orDCH.

tan =

H

E D

A C

B

H

E D G

A C

B

10 cm

5 cm

5 cmF


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Guided Exercise:

GuidedSolution:

(a) Using Pythagoras Theorem,DE= 12 cm,EG = 15 cm

EH= 22 158 +

= ..

(b) Step 1: Step 2:

Identify the angle between the two planes

(EHIandDGHI)

Step 3: Step 4:

Sketch the right angled triangle Using Trigonometry Ratio to

in Step 2. calculate the answer.

8 cm

The figure shows a right prism with a

uniform trapeziumDEJIas its cross section.Calculate:

(a) the length ofEH,

(b) the angle between the planes

EHIandDGHI.

6 cm

15 cm

H

E

H

8 cm

G

II

G

D E

17 cm

DIE

D E

tan DIE=8

12

= 1.5

DIE = 56.32o or 56o19

I

D E

DIE

H K

10 cmI J

G F

9 cm

D E12 cm


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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.

A N B

D M C Line of intersection isXOY.

Lines perpendicular toXOY

are XD and XA, OMand ON

orYCand YB.X O Y

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

Plane 2

Plane 1

line ofintersection

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