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Introduction to Geometry
Motivation: In our daily life, many articles are with different shapes. But what are the differences between those shapes? We know some of them are different; but are there any common properties between those different shapes? Can we find any rules from those different things?
The study of Geometry is with long history. At very beginning, it was focusing on survey to measure the area or the distance in a very large scale. Along with that, the other branches of science were also developed – like trajectory of moving articles, study for the movement of the heavenly bodies, … And those triggered more study on the other fields. To understand how our civilization is established, it would be better to start from those very fundamental things. Some basic terms on geometry will be introduced here and we will put more emphasis on their mathematical meanings.
Point Mathematically when we speak of a “point”, it stands for a position. And it is with no length, width, or volume. Usually, we use a letter to represent a point, for example, point A, or point P.
Line Segment Line segment is defined as the shortest path between two points. And the two points are known as Endpoints of the Line Segment. For two points A and B, the line segment with endpoints A and B is denoted as . This notation is also used to represent the length of the line segment.
You might feel strange why we define those terms in this way. When we were very young, we were told to use ruler to link two points as a line segment. But we now realize that ruler might not be straight enough. How “straight” is straight ? We do not have answer for that in real world. Some people used to say that the path that light travels is straight. But that might not be always true according to our current knowledge.
Line We just use the concept about “straight” to describe the definition of a line. Mathematically, a line is a line segment extends infinitely on both directions. For a line passing two points A and B, we denote it as .
Ray Ray is defined as line segment extends only in one direction. And the point where the ray begins is the endpoint of the ray. For example, “ray AB” ( denoted as ) is starting from point A and passing B to extends infinitely.
Intersection of Two Lines
If a point is both on two lines, then the point is the intersection of the two lines.
And in the figure above, we say that two lines “intersect” at point E. Please notice that the two lines can uniquely determine a plane.
Plane We define “plane” from two lines intersecting at one point. The plane is the collection of all the points such that every point can be passed via a straight line determined by selecting two points respectively from the original two lines intersecting at one point.
Conceptually, you can think a plane as a “flat” sheet without thickness. But how “flat” is flat? That’s why we would like to start from two intersecting lines. As a matter of fact, there are many other ways to determine a plane:
1. three distinct points 2. a line and a point that is not on the line 3. two parallel lines
Parallel Lines Two lines in the same plane are said to be parallel to each other if they have no intersection. Please notice that we require that the two lines must be on the same plane. If line L is parallel to line M, we use L // M to represent this relationship.
Angle Two rays sharing the same endpoint form an angle. The endpoint is known as the vertex of the angle. We use the notation to represent an angle. When we specify an angle, we use its vertex to call that angle. If the vertex is shared by many angles, we use 3 letters to specify that angle to avoid confusion. Please check the following diagrams:
A or BAC . When we use 3 letters to specify an angle, the second letter is always the vertex of that angle. For a line, we specify that its angle is 1800 ( pronounced as “180 degree” ). “Degree” is the unit we use to measure angle. Sometimes the notation BAC is also used as the magnitude of the angle.
Right Angle If an angle is 900, we call this angle as “right angle”.
Perpendicular Lines If two lines meet at right angle ( 900 ), we say that the two line are perpendicular to each other. Usually, in the diagrams, we use a small box to indicate that angle is 90o as shown below. And we use the notation “ L M “ to indicate the two lines L and M are perpendicular to each other.
Triangle
A triangle is a closed area formed by three joint line segments with 3 points as sharing endpoints shown as below. We use ABC to denote it. Sometimes, we also use ABC to stand for the area of the triangle.
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