Chapter 11 Lines and Planes in 3-Dimensions

8/3/2019 Chapter 11 Lines and Planes in 3-Dimensions

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11.1 ANGLE BETWEEN LINES AND PLANES

A Differentiating Between 2-D and 3-D Shapes

2-D Shapes 3-D Shapes

Have length and breadth

only

Have area only

Examples:

Have length, breadth andheight or depth

Have both area and volume

Examples:

Square Triangle Cuboid Cylinder


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B Identifying Horizontal Planes, Vertical Planes

And Inclined Planes

1 A plane is a flat surface.

2 There are three types of planes, namely, horizontal

plane, vertical plane and inclined plane.

Horizontal plane

Vertical plane


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Inclined plane

B Identifying Horizontal Planes, Vertical Planes

And Inclined Planes


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C Sketching A 3-D Shape and Identifying

The Specific Planes

X

L

K

O

M

N

Plane Type

KLMN

XON

LMX

XKM

XNK

Horizontal

Vertical

Inclined

Vertical

Inclined


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A B

CD

D Identifying Lines That Lie or Intersect With

A Plane

R

S

In the diagram above, the line RS said to lie on the plane ABCD.

Each point on the line RS lies on the plane ABCD.


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A B

CD


A Plane G

H

In the diagram above, the line GH said to intersect with the plane

ABCD. The line GH meets the plane ABCD at the point only.


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A B

CD


A Plane

X

In the diagram above, the line AX and BX are also

intersect with the plane ABCD.

X


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A B

CD

E Identifying Normals to A Plane

A normal is a straight line that is perpendicularto a plane.

Normal to plane


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Orthogonal projection on plane

A B

CD

O Q

P

Normal to plane

F Drawing and Naming The Orthogonal Projection

of A Line on A Plane

The orthogonal projection of a line OP on a plane with a point O

is the line OQ. Q is the point of intersection of the normal from

P to the plane.


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of The Line GK on The Plane JKLM

Line Orthogonal

Projection

GK KM

D

E

F

G

J

K

L

M


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of The Line GK on The Plane DEFG

Line Orthogonal

Projection

GK GE

D

E

F

G

J

K

L

M


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of The Line MS on The Plane PKNS

Line Orthogonal

Projection

MS SN

P

Q R

S

K

L M

N


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of The Line PL on The Plane QLMR

Line Orthogonal

Projection

PL LQ

P

Q R

S

K

L M

N


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of The Line NL on The Plane PQLK

Line Orthogonal

Projection

NL LK

P

Q

R

SK

L

M

N


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of The Line LN on The Plane RMNS

Line Orthogonal

Projection

LN NM

P

Q

R

SK

L

M

N


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Orthogonal projection on plane

A B

CD

O Q

P

Normal to plane

G Determining The Angle Between A Line

and A Plane

The angle between a line and a plane is the angle between the line

and its orthogonal projection on the plane.

< POQ is the angle between the line OP and the plane.


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D

E

F

G

J

K

L

M


and A Plane

Line

Plane

Orthogonal

projection

Angle

JKLM

GK

KM

< GKM


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D

E

F

G

J

K

L

M


and A Plane

Line

Plane

Orthogonal

projection

Angle

DEFG

KG

GE

< KGE


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and A Plane

Line

Plane

Orthogonal

projection

Angle

RMNS

LN

NM

< LNM

P

Q

R

SK

L

M

N


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and A Plane

Line

Plane

Orthogonal

projection

Angle

PKNS

MS

SN

< MSN

P

Q R

S

K

L M

N


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11.2 ANGLE BETWEEN TWO PLANES

A Identifying The Line of Intersection BetweenTwo Planes

1 The planes that intersect meet at a straight line.

2 The line is called the line of intersection of the

two planes.

Line of intersection


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S

C

R U

B

T

P

A

Q

B Determining The Line On Each Plane Which IsPerpendicular to The Line of Intersection of The

Two Planes The straight line

AC is on the plane

PQRS and the

straight line BC is

on the plane RSTU.


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S

C

R U

B

T

P

A

Q


Two Planes Both the lines AC

and BC are drawn

from the point C,

which is on the line

of intersection

between plane PQRS

and plane RSTU.


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S

C

R U

B

T

P

A

Q


Two Planes Both the lines AC and

BC are perpendicular

to the intersection

between the

two planes.


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S

C

R U

B

T

P

A

Q


Two Planes The edges QR and UR are

also straight lines, with QR on

the plane PQRS and UR on

the plane RSTU. Both lines

are drawn from the point R,

which is on the line of

intersection of thetwo planes are

perpendicular to it.


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S

C

R U

B

T

P

A

Q


Two Planes The edges QR and UR are

also straight lines, with QR on

the plane PQRS and UR on

the plane RSTU. Both lines

are drawn from the point R,

which is on the line of

intersection of thetwo planes are

perpendicular to it.


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B

X

ED

N

C

A

M

F

C Determining The Angle Between Two Planes

The angle between twointersecting planes is the

angle between two lines,

on each plane. These two

lines must have a common

point and perpendicularto

the line ofintersection

between the two planes.

Angle between

Two planes

< MXN is the angle between the planeABEF

and the plane BCDE.


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EXAMPLEState the angle between the plane PQLM

and the plane PQKN

P Q

RS

K

L M

N

The angle between the planes

PQLM and PQKN

< LPK or < MQN


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D G

M

LK

FJ

E

EXAMPLEState the angle between the plane GJK

and the plane JKLM

< GJM


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D G

M

LK

FJ

E

EXAMPLEState the angle between the plane GJK

and the plane DEJK

< GJD


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TRIGONOMETRIC RATIO

ADJACENT SIDE (A)OPP

OSITESID

E

(O)

Sin = O

HCos = A

H

Tan = OA


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PYTHOGORAS THEOREM

a

b a2 + b2 = c2


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PYTHAGOREAN TRIPLES

32 + 42 = 52

62 + 82 = 102

92 + 122 = 152

52 + 122 = 132

82 + 152 = 172

3 4 5

6 8 10

9 12 15

5 12 13

8 15 17

Red colorednumber is the hypotenuse


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6

108 15

9

12

13

12

5

17

8

15

?

?

?

?


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UT

C

9 cm

BA

D

R

QP

S

15 cm

12 cm

Calculate the angle between plane PBCand planeABCD.

9 cm

12 cm

Tan PUT = 129

PUT = 5308

Line of intersection

Line that is perpendicular

to the line of intersection

Normal

Orthogonal

projection


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UT

C

9 cm

BA

D

R

QP

S

15 cm

12 cm

Calculate the angle between plane PBCand plane BCRQ.

9 cm

12 cm

Tan PUQ = 9

12 PUQ = 36052


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R

Q

S

P

T U

VW

5 cm

12 cm

4 cm

Calculate the angle between plane PRV and plane QRVU.

12 cm

5 cm

Tan PRQ = 12

5 PRQ = 67023


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T

U

V

W

S

R

10 cm12 cm

Calculate the angle between plane SRVand plane RSTU.

5cm

5cm

12 cm

Tan VST = 5

12 VST = 22037


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R

Q

S

P

T U

VW

5 cm

12 cm

4 cm

DIAGRAM 2

Diagram 2 shows a cuboid with base TUVW.

Calculate the angle between plane PRV and plane RSVW.

12 cm Tan PRS = 5

12 PRS = 22037


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Q

R

S

P

E

D

G

F

J

Diagram 5 shows a cuboid PQRSDEFG with a horizontal square base PQRS.

J is the midpoint ofDG. QR= RS= 12 cm and FR= 8cm.

(a) Name the angle between the plane JRSand the plane RSGF,

(b) Calculate the angle between the plane JRSand the plane RSGF.

DIAGRAM 5

8 cm12 cm

6 cm

8 cm (b) Tan JSG = 68

JSG = 36052

(a) JSG P1

K1

N1


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yW

13 cm

10 cm 12 cm

P Q

R

S T

U

Diagram 6 shows a right prism with an isosceles triangle base, STU. The

isosceles triangle STU is the uniform cross-section of the prism.

ST = SU and W is the midpoint of TU.

(a) Name the angle between the line PWa

the base STU,

(b) Calculate the angle between the line P

the base STU.

8 cm

(b) Tan PWS = 138

PWS = 58024DIAGRAM 6

(a) PWS P1

K1

N1


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T

U

W

V

P

Q

R

S

M

N

y

y8 cm

6 cm

12 cm

5 cm

DIAGRAM 7

Diagram 7 shows a right prism with a horizontal rectangular base PQRS.

VUQR is a trapezium. Mand Nare the midpoints ofPSand QRrespectively.

(a) Name the angle between the lineTRand the base PQR

S,

(b) Calculate the angle between the line TR and the base PQRS.

8 cm

6 cm

10 cm

5cm

(b) Tan TRM = 510

TRM = 26034

(a)

TRM

P1

K1

N1


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8 cm

5 cm

9 cm

12 cm

P

QR

U

W

T

E

F

S

V

Diagram 9 shows a right prism with a horizontal rectangular base PQRS.

Given that Eand Fare midpoints ofWVand SRrespectively.

a) Find the length ofPF

b) Calculate the angle between the line PEand the plane PQRS

c) Name the angle between the plane PQVEand the plane PQUT.

DIAGRAM 9


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8 cm

5 cm

9 cm

12 cm

P

QR

U

W

T

E

F

S

V

a) Find the length ofPF

8 cm

6 cm10 cm

10 cm


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8 cm

5 cm

9 cm

12 cm

P

QR

U

W

T

E

F

S

V

b) Calculate the angle between the line PEand the plane PQRS.

9 cm

10 cm

Tan E

PF

= 910 EPF = 41059


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8 cm

5 cm

9 cm

12 cm

P

QR

U

W

T

E

F

V

S

c) Name the angle between the plane PQVEand the plane PQUT.

< VQU


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Diagram 1 shows a right prism with a horizontal square base HJKL.

Trapezium EFLK is the uniform cross-section of the prism. Therectangular surface DEKJ is vertical while the rectangular surface GFLH

is inclined. [4 marks]

6 cm

8 cm

D

E F

G

J

KL

H

Diagram 1

Calculate the angle between

the plane DLH and the base

HJKL.

SPM 2003 : No 4


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6 cm

8 cm

D

E

G

J

K L

H

Answer:

8 cm

6 cm 10cm

F

< DHJ


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D

HJ

Tan DHJ = 68

DHJ = 36.90

P1

K1

N1

6

8

10


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ANOTHERANSWERS

Sin DHJ = 610

DHJ = 36.90

Cos DHJ = 8

10 DHJ = 36.90

Documents

Chapter 11 Lines and Planes in 3-Dimensions