GEOMETRY
The standard Competences:
Determine the position, distance, and angle that include point, line and plane in the three dimension space.
The Basic Competence:
> Determine the position of point, line and plane in the 3 dimension space
> Determine the distance from point to line and from point to plane
> Determine the size of angle, between line and plane and between two plane
1. POSITION OF POINT, LINE AND PLANE IN SPACE
Point, line and space are parts of the three dimension space. These part are named as space elements. In this chapter, we will study about position of point, line and plane in space. The beginning of this study is discussed about the meaning of point, line and plane, along with the axioms and theorems that relevant with them.
1.1 The Meaning of Point, Line and Plane
Space elements (point, line ,plane) are base term in geometry. As we had known that base term is terminology that only can be described in the following explain.
- Point
Point only can be identified by the place, but it doesn’t have size(has no dimension). Point is shown by dot sign, then follow with the name of that point. The name of that point usually use capital letter, like A,B,C,P,Q and R.
- Line
We can lengthen the line(means straight line ) as long as we want. But remember limit of the place that we use to draw the line, a line is drawn by just the part of line. This part of the line is called representative line. Line just has the size of long, and doesn’t have the size of wide. Name of a line can be identified by the name of representative line with use small letter, like g,h,k or say the name of segment line from starting point to the end of point.
- Plane
Plane can be expanded. Generally a plane just drawn by the representative plane. The representative plane has two size, long and wide. Figure from plane has shape like square or rhombus, rectangle or parallelogram. The name of representative plane is written in the corner of the plane and use character a,b,c or H,U,W or with mention angle points from that plane.
- Axioms of line and plane
Beside of the base terms, study geometry need axioms (usually called postulate). Axiom is statement that consider truth in a system and that truth receive without proof. In space of geometry, there are three important axioms. Those axioms are introduced by Euclides (300BC), he is mathematician from
These are the Euclides’s axioms :
Axiom 1 : Across two any kind of points, it can be made a straight line.
Axiom 2 : If a line and a plane has two alliance points, then that line is on plane.
Axiom 3 : across three any kind of points we can make a plane.
Notes:
1. In axiom 1, two kind of points are that two points are not same
2. In axiom 3, three kind of points are that points are not in one line.
According to the three axioms, then it can be 4 theorems to determine a plane.
Theorem 1: a plane is determined by three any kind of points
Theorem 2: a plane is determined by a line and a point (the point outside the line).
Theorem 3: a plane is determined by two intersection lines
Theorem 4: a plane is determined by two parallel lines.
Base term about point, line and plane, then axioms and theorems about line and plane are already known. Then we will discuss about position of point, line and plane in three dimension space.
1.2 Position of Point to Line and Point to Plane
- Position of point to line
> The point is in the line
If line g across point A, then point A is in line g
> The point is in outside the line
If line h does not across point B, then point B is not on line h (outside the line g)
Example, look at cube ABCD-EFGH.
a) The vertect of cube that on line g are point A and B
b) The vertect of cube that in outside line g are point C,D,E,F,G, and H.
- Position of Point to Plane
> The point is on plane
If point A can be acrossed by plane a, then point A is on plane a.
> the point is in outside plane
If point B cannot be acrossed by plane b, then point B is in outside plane b
Example, look at cube ABCD-EFGH. Plane DCGH as the representative of plane U.
a) The vertect of cube that on plane U are point C,D,G and H.
b) The vertect of cube that in outside of plane U are point A,B,F and E.
Exercise 1.
1. Use the top of index finger as point A and pencil as line g.
a) Display the position of point A , if point A is on line g!
b) Display the position of point A , if point A in outside line g!
c) Is there any position of point A to another line g?
2. There is a cube
a) on line k
b) in the outside of line k
c) on line l
d) in the outside of line l
e) online m
f) in the outside of line m
3. Use the top of index finger as point A and notebook as plane U.
a) Display the position of point A , if point A is on plane U!
b) Display the position of point A , if point A in outside of plane U!
c) Is there any position of point A to another plane U?
4. There is a regular pyramid T.ABCD
a) Mention the vertect of pyramid that are located in the flanks of side.
b) Mention the vertect of pyramid that are located in the flanks of base.
c) Mention the vertect of pyramid that are located in the planes of side.
d) Mention the vertect of pyramid that are located in the base of plane.
e) Is there any vertect of the pyramid that in the outside of flanks
f) Mention the agle of pyramid that in the outside of the base flank
g) Mention the agle of pyramid that in the outside of base plane
5. There is a cube KLMN-PQRS, plane KLMN represent plane a, plane KLQP represent plane b and plane KMRP represent plane c. Mention the vertect of that cube that:
a) are located on plane a
b) are located in the outside of plane a
c) are located on plane b
d) are located in the outside of plane b
e) are located in plane c
f) are located in the outside of plane c
1.3 Position of Line to Line and Line to Plane
A. Position of Line to Another Line
The position possibility of a line to another line in three dimension space are intersect, parallel or cross.
> Two intersect lines
Line g intersect to line h if line g and line h are located in a plane (remember theorem 3) and have alliance point. In geometry of plane, the alliance point is called the intersection point between two lines.
Notes:
If two lines are intersect in more than one point, so that two line are same.
> Two parallel lines
Line g and line h are parallel lines, if line g and line h are located in a plane (remember theorem 4) and have not any alliance point
> Two across lines
Line g across (not intersect and not parallel) to line h, if line g and line h are not located in a same plane.
Example :
Look at the cube ABCD-EFGH
a) The sides of that cube that intersect to line g are AD,
b) The sides of that cube that parallel to line g are BC, EF and HG.
c) The sides of that cube that across to line g are CG ,DH, EH and FG.
d) Is there any side of cube that same with line g?
> Axiom of two parallel lines :
Axiom 4 :
Across a point that is located in the outside a line , it only can be made a line that parallel with that line.
> Theorems of two parallel lines
According to the position of two parallel lines, it can be pulled same conclusion. The conclusions are described in these theorems:
- Theorem 5 :
If line k parallel to line l, and line l parallel to line m, so line k parallel to line m.
- Theorem 6 :
If line k parallel to line h and intersect line g, line l parallel to line h and intersect line g too, so line k, l and g are located in a plane.
- Theorem 7 :
If line k parallel to line l, and line l pierce plane a, so line k pierce plane a.
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