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Points and Lines

Points

A point indicates position, it is represented by a dot and named by a capital letter. The Figure below illustrates point A, point B, and point C.

what is a point? how to define a point?

Lines

A line (straight line) is a set of continuous points that extend indefinitely in either direction. It is named by any two points on the line. The symbol ↔ written on top of two letters is used to denote the line. A line may also be named by one small letter. The Figure below illustrates one line can be written as, line AB, or line m.

what is a line? how to determine a line?

Postulate 1.1: A line contains at least two points.

Planes

A plane is a set of points that forms a flat surface which extends indefinitely in all directions. It is usually represented as a closed four sides figure. A capital letter is placed at one of the vertex. The Figure below illustrates the plane P and plane Q.

what is a plane? how to determine a plane?

Line Segments

A line segment is a part of a line. It has two endpoints and is named by its endpoints. Sometimes, the symbol   ̅ written on top of two letters is used to denote the segment. The Figure below showed the line segment AB.

what is a line segment? how to determine a line segment?

Rays

A ray is a part of a line, except that it has only one endpoint and continues forever in one direction. It is named by the letter of its endpoint and any other point on the ray. The symbol → written on top of the two letters is used to denote the ray. Figure below shows ray AB.

what is a ray? how to determine a ray?

Midpoint

A midpoint of a line is the point that has equidistant from the two endpoints. See Figure below, C is the midpoint of line segment AB because the length of AC is the same as the length of CB, that is, AC = CB or because AC is equal to one-half AB and CB is equal to one-half AB.

what is a midpoint? how to determine a midpoint?

Collinear Points

Collinear points are points which lie on the same line. See Figure below, points A, C, and B are collinear because they lie on the same line.

what is the collinear points?

See Figure below, points E, G, F are not collinear because point G do not lie on the line EF.

Two Points Determine one Line

Through any two points, there is exactly one line. See Figure below, these is only one line l that pass through point A and point B.

how many points determine a line?

Two Lines Intersect

If two lines intersect, then they intersects in exactly one point. See Figure below, both line m and line n intersect at point P.

when two lines intersect, what happends?

The Shortest Distance between Two Points

Of all connections between two endpoints, the straight line has the shortest distance. See Figure below, both the line segment l and the curve m have the same endpoints, but the length of the straight line l is less than the length of the curve m.

what is the shorted distance between two points?

The straight line segment AB is the shortest distance from point A to point B.

Distance of a Line Segment

he distance between two points is the length of the line segment that join the two endpoints. See Figure below, the letter d denote the distance from A to B. The length of the segment AB can be written as d = AB.

how to measure the distance of a line segment?

Note: the term “line” will always be understood to represent a straight line. The term “segment” represents a part of a line.

Point Lies between Two Endpoints

If point P lies between points A and B on a line, then AP + PB = AB. See Figure below, since points A, P, and B are collinear, and point P lies between points A and B, then AP + PB = AB.

what is the total length of a line segment in which three points lie on it?

More Than One Plane Contain Two Points

A plane contains at least three not collinear points. See Figure below, both plane P and plane Q contain point A and point B. More than one plane contain two points.

Are two points determine a plane?

Three Not CollinearPpoints Determine a Plane

Through any three not collinear points, there is exactly one plane. See Figure below, point A, point B, and point C lie on plane P.

how many points can determine a plane?

Two Planes Intersect

If two plane intersect, then their intersection is a line. See Figure below, plane P and plane Q intersect along line l.

what happends when two plane intersect?

Line and a Point Out of the Line Determine a Plane

If a point lies outside a line, then exactly one plane contains both the line and the point. See Figure below, the plane P contains line l and point A.

how to determine a plane?

Two Intersect Lines Form A Plane

If two lines intersect, then exactly one plane contains both lines. See Figure below, the plane Q contains line m and line n, which intersect at point A.

how many ways to determine a plane?

Coordinate of Points

Each point on a line can be paired with exactly one real number called its coordinate. The distance between two points is the positive difference of their coordinates.

how to determine the coordinate of a point?

See Figure above, the coordinate of point B is b, the coordinate of point A is a. Since the line has only one direction, then b > a, then the distance of the segment AB is equal to the coordinate of B subtract the coordinate of A. The distance AB = b – a.

Example 1

Find the length of BH.

example 1 of points on aline and find the distance between two points.

Solution
Since B = 3 and H = 15
So BH = 15 - 3 = 12

Example 2

Find RS = ? If ST = 7 and RT = 15

example 2 of points on aline and find the distance between two points.

Solution
Since point S lies between points R and T, so RS + ST = RT
So RS = RT - ST = 15 - 7 = 8

Example 3

Given: C is the midpoint of AB. D is the midpoint of AC. If AB = 1, find DC = ?

example 3 of points on aline and find the distance between two points.

Solution
Since C is the midpoint of AB, so AC = CB = AB/2 = 1/2
Since D is the midpoint of AC, so AD = DC = AC/2 = (1/2) ÷ 2 = 1/4