OLEH

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LINES AND PLANES IN THREE DIMENSIONS
OLEH PN. CHUANG BEE ENG SM TEKNIK BATU PAHAT
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LINES AND PLANES IN 3-DIMENSION

• To answer the question from this topic, the
  students must acquire the following skills
• Able to identify the angle between a line and a
  plane ( 1 Mark )
• Able to calculate the angle between a line and a
  plane ( 3 Marks ).
• Able to identify the angle between two planes ( 1
  Mark ).
• Able to calculate the angle between two planes (
  3 Marks )

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EXAMPLE
Diagram shows a cuboid with a horizontal
rectangular base. Calculate the angle between
the planeTWR and the plane PSWT.
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ACTIVITY 1 TO IDENTIFY THE PLANE
G
H
E
F
D
C
A
B
PLANE AT THE TOP
PLANE EFGH
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G
H
E
F
D
C
A
B
PLANE ON THE LEFT
PLANE ADHE
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G
H
E
F
D
C
A
B
PLANE IN THE FRONT
PLANE ABFE
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G
H
E
F
C
D
A
B
PLANE AT THE BACK
PLANE DCGH
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G
H
E
F
C
D
A
B
PLANE AT THE BOTTOM
PLANE ABCD
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G
H
E
F
C
D
A
B
PLANE ON THE RIGHT
PLANE BCGF
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THE LOCATION OF THE POINT
ON TOP OF THE RED DOT
TO THE RIGHT OF THE RED DOT
AT THE BACK OF THE RED DOT
IN FRONT OF THE RED DOT
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ON THE TOP OF .
AT THE BACK OF .
TO THE LEFT OF .
TO THE RIGHT OF .
IN FRONT OF .
AT THE BOTTOM OF .
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ACTIVITY 2 TO DETERMINE THE LOCATION OF A POINT
POINT E
POINT TO THE LEFT OF F

POINT AT THE BOTTOM OF F
POINT B
POINT G
POINT AT THE BACK OF F
POINT TO THE RIGHT OF D
POINT C
POINT ON TOP OF D
POINT H
POINT IN FRONT OF D
POINT A
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ANGLE BETWEEN A LINE AND A PLANE
A
LINE
PLANE
B
C
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Activity 3 To Identify The Angle Between Line
And Plane
H
G
The line draw from G and perpendicular to the
plane ABCD is call normal
Normal
E
F
D
C
The line lies on the plane ABCD which joint the
point A to the line GC is known as the orthogonal
projection of line AG on the plane ABCD.
Orthogonal projection
A
B
The angle between the line AG and the orthogonal
projection, AC is the angle between the line AG
and the plane ABCD that is GAC.
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ACTIVITY 3 To Identify The Angle Between A
Line And A Plane
Example 1a
H
G
C
A
G
Normal
E
F
D
At the bottom
C
Name the angle between the line AG and the plane
ABCD
A
B
Orthogonal projection
Angle between the line AG and the plane ABCD
GAC.
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EXAMPLE 1(b)
Diagram 1b shows a cuboid ABCDEFGH. Name the
angle between the line HB and the plane ABCD.
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ACTIVITY 4 To find the angle between a line
and a plane
Example 2(a)
Diagram 2(a) shows a cuboid, ABCDEFG. Find the
angle between the line AH and the plane DCGH.
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Solutions
Steps
No
Draw the line AH and shade the plan DCGH in
diagram 2a.
1.
12cm
H
G
5cm
E
F
D
C
4cm
A
B
Diagram 2a
Diagram 2a shows a cuboid, ABCDEFG. Find the
angle between the line AH and the plane DCGH.
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No Steeps Solutions
2 Use the method you have learned in activity 3, identify the angle between the line AH and the plane DCGH
H
A
D
back
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No Steps Solutions
3 Refer to the points you have obtained in steep 2 (point A, H, D), complete the ? AHD. Mark AHD. Mark the right angle, HDA. Transfer out the ? AHD.
H
D
A
12cm
H
G
5cm
E
F
D
C
4cm
A
B
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No Steps Solutions
4 With the information given in the question, label the length of the sides of ? AHD. At least the length for 2 sides must be known. Use Pythegoras Theorem if necessary.
H
5 cm
A
D
4 cm
12cm
H
G
5cm
E
F
D
C
4cm
A
B
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No Steps Solutions
6
H
Mark, - the opposite side, AD asT - the
adjacent side, HD as S
5 cm
S
D
A
4 cm
T
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No Steps Solutions
6 Use the tangent formula to calculate AHD. Remember, use The sine formula, if O and H were known - The cosine formula, if A and H were known The tangent formula, if O and A were known Tan AHD AHD tan -1 AHD 38040
- SOH
CAH
TOA
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example 2 (b)
12 cm
G
H
E
F
4 cm
D
C
3 cm
A
B
Diagram 2b
Diagram 2b shows a cuboid,ABCDEFGH. Calculate the
angle between the line HB and the plane BCGF
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ANGLE BETWEEN TWO PLANES
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ACTIVITY 5 To Identified The Angle Between Two
Planes
EXAMPLE 3(a)
G
H
E
F
D
C

1. DRAW 3 LINES

A
B
Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane AGH and the plane ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
E
F
D
C
Bottom
A
B

1. Mark the location (direction) of the plane ABCD
   at the bottom of the first line to the left.

Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane, AGH and the plane, ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
A
E
F
D
C
Bottom
A

1. Refer to the plane, AGH, identify the points
   which touch the plane, ABCD and write it at the
   middle line.

B
Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane, AGH and the plane, ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
H / G
A
E
F
D
Bottom
C

1. Refer to the plane, AGH, identify the point which
   does not touch the plane, ABCD and write it at
   the first line to the left.

A
B
Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane, AGH and the plane, ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
A
H/G
Bottom
E
F
D
C

1. Between the point H and G, point which is nearer
   to point A or located on the same plane as point
   A will be choosen. Point which is not choosen
   will be earased.

A
B
Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane, AGH and the plane, ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
A
H
Ke Bawah
E
F
D
C

1. Between the point H and G, point which is nearer
   to point A or located on the same plane as point
   A will be choosen. Point which is not choosen
   will be earased.

A
B
Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane, AGH and the plane, ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
A
H
D
E
F
D
C

1. Identify the point which is located at the bottom
   of the point H ( )and write it on the first
   line to the right.

A
B
Diagram 3a
Diagram 3a shows a cuboid, ABCDEFGH. Name the
angle between the plane, AGH and the plane, ABCD
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ACTIVITY 5 To Identified The Angle Between Two
Planes
G
H
E
F
D
C
Bottom

• In the diagram 3a, complete the ? HAD and mark
  the HAD

A
B
Diagram 3a
Angle between the plane, AGH and the plane,
ABCD HAD
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EXAMPLE 3(b)
Diagram 3b shows a cuboid with horizontal
rectangle base ABCD. Name the angle between the
plane ACH and the plane CDHG
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THANK YOU