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GEOMETRIC
POSITION OF A LINE TOWARD A PLANE
POSITION OF A LINE TOWARD A PLANE
Kinds of possible position of a line toward other line in a plane:
1)
h
g
Line g and line h is intersected
2)
g
h
line g and line h is parallel
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POSITION OF A LINE TOWARDS OTHER LINES
3)
g
In a plane of a there is line g, then line h intersects plane a
and line h doesn’t have point of intersection with line g.
line g and line h is crossed over
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Axioms of Two Parallel lines
axiom 4
h
A
g
Through a point outside the line, we only can make a
line that parallel with the line.
in the figure above, point A is outside line g. Through point
A and line g, we can make a plane a (look at Rule number 2,
a plane is determined by a point and a line). Next, through
point A, we can make line h which parallel with line g.
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Rules of Two Parallel Lines
Rule number 5
k
Line k parallel with line l
l
Line l parallel with line m
m
Then line k is parallel with line m
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Position of a Line Toward a Plane
 Rule number 6
h
line k parallel with line h
k
line k intersect line g
l
line l parallel with line h
g
also intersect line g
Then, lines k, l, and g are in a plane
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Position of a Line Towards a Plane
Line k parallel with line l
k
Line l intersect plane a
l
Then line k intersect plane
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Position of a Line Towards a Plane
1)
g
A
B
Line g is in plane a if line g and plane a at least have 2 points
of intersection
(based on axiom 2, if a line and a plane have 2 points of
intersection , then the whole line is in the plane)
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Position of a Line Towards a Plane
h
Does line h parallel with plane α ?
Line h is parallel with plane a , if line h and plane a doesn’t
have any points of intersection.
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Position of a Line Towards a Plane
k
Does Line k intersect plane α ??
Line k intersect plane α, if line k and plane α only have a
point of intersection.
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Example:
1.
Let Cube ABCD EFGH
H
E
g
G
F
D
A
C
B
The edge of AB as the representative of line g.
» The cube edges that intersect with line g is.......
(AD, AE, BC, and BF)
» The cube edges that parallel with line g is....
(DC, EF,dan HG).
» The cube edges that cross over line g is.....
(CG, DH, EH, and FG).
» Is there any cube edge that parallel with line g?
(AB)
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2. Given cube
H
E
G
F
D
A



U
C
B
The cube edges that in plane U is.....
(AB, AD, BC, dan CD).
The cube edges that parallel with plane U is.....
(EF, EH, FG, and GH).
The cube edges that intersect plane U is....
(EA, FB, GC, and HD).
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Rules About lines Parallel with Plane
Rules number 8
g
h
If line g parallel with line h and line h is on plane a ,
then line g parallel with plane a .
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POSITION OF A LINE TOWARD OTHER LINES
 Rule number 9
g
If plane a through line g and line g is
parallel with plane β, then the intersection
line of plane a and plane β will be parallel to
line g
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Rule number 10
g
h
a,
If line g parallel with line h and line h parallel to
plane a, then line g is parallel to plane a
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POSITION OF A LINE TOWARDS OTHER LINES
Rule number 11
(a , β)
If plane a and plane β intersected and each of them
parallel to line g, then the intersection line between
plane a and plane β will be parallel with line g.
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POSITION OF A LINE TOWARDS OTHER LINES
Note: in rules number 9 and 11 need
concept of intersection line between
two planes.
The concept of intersection line
between two planes will be discussed
in the next meeting.
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Angle and Plane in Drawing Polyhedral
A.
The Intersection of Line with Plane
If there is a line and a point in a plane, then there will be
3 possibilities:
1. The line is in the plane if all points in the line is in that
plane.
2. The line parallel with plane, if there is no point of
intersection between line and plane.
3. The line intersected the plane, if it only has one point
of intersection between line and plane.
B.
Distance Between Points and Plane
The distance of a point to a plane is the distance of this
point to its plane projection.
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Angle and Plane in Drawing Polyhedral
c. Angle Between Line and Plane
The angle between line and plane is an angle
between the line and its projection in a plane.
D. Angle Between Two Planes
Angles of two planes that intersected in line AB is an
angle between two lines in a plane. Each of them are
perpendicular to plane AB and intersect in one point.
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Distances in Polyhedral
1.
Given a cube ABCD.EFGH with edge length 8 cm. Points P,Q and R are
in the mid points of edges AB,BC and plane ADHE respectively .
Find the distance between:
a. Points P and R
b. Points Q and R
c. Point H and line AC
Answer :
a. See that ∆PAR has a right angle on A
H
G
AP = ½AB = 4 cm
AR = ½AH =½ AD 2  DH 2
E
F
= 1 82  82  4 2
2
R•
PR =
D
C
•Q
S•
A
P•
B
=
=
AP2  AR2
42  (4 2)
2
48  4 3
So, the distance points P and R is 4 3cm
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Angle Formed by line and a plane
E
A
Example.
Given a cube ABCD.EFGH with edge length 10 cm.
a. Draw an angle between line AG and plane ABCD.
b. Measure the angle size.
Answer :
G
H
a. Projection of line AG onto plane ABCD is line
F
AC So, the angle between line AG and plane
ABCD is GAC = a
Da
C
b. See that CG = 10 cm and AC= 10 2 cm
because AC is the diagonal of cube’s face.
B
See that GAC has a right angle on C, then
tan a = CG  10  1 2
or
a =35,30
AC
10 2
2
Then, the angle size between line AG and plane ABCD is
a = 35,30
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Example:
Given cube ABCD.EFGH with edge length a single.
Draw and find the angle between plane BDE and BDG
Answer:
Look at the following figure. The angle between plane BDE and
plane BDG is
a. See that ∆EPA is right angle in A,so that..
a
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The End
See You
Next Meeting
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