1. Introduction to Basic Geometrical Ideas
Geometry starts with a few basic ideas: points, lines, rays and line segments. These are like the alphabet of geometry. We use them to build shapes, draw angles and understand diagrams. Once these words are clear, the rest of geometry becomes much easier to follow.
2. Points and Lines
In this part we look at what a point and a line mean in geometry, how we draw them and how we name them using capital letters.
2.1. Points
A point shows an exact position or location. It is usually drawn as a small dot. A point has no length, no width and no thickness. We name a point using a capital letter like \( A \), \( B \) or \( P \).
For example, when we mark a corner of a book or a spot on a map, we are thinking of a point in geometry.
2.2. Lines
A line is a straight path that extends endlessly in both directions. A line has no endpoints and no fixed length. We imagine it going on forever on both sides, even though on paper we can draw only a small part of it.
We usually name a line using two points on it. For example, a line passing through points \( A \) and \( B \) can be written as line \( AB \) or as \( \overleftrightarrow{AB} \).
On a diagram it may look like a long straight stroke with arrow marks on both ends to show that it continues.
3. Line Segments and Rays
Lines go on forever, but in daily life we mostly deal with parts of lines. Two important parts of a line are line segments and rays. Both come from a line but have different end behaviour.
3.1. Line Segments
A line segment is a part of a line that has two fixed endpoints. Because both ends are fixed, a line segment has a definite length.
If the endpoints are \( A \) and \( B \), the line segment is written as \( \overline{AB} \). On paper we draw it as a straight stroke from point \( A \) to point \( B \) without arrows.
Examples: the edge of a notebook, the side of a table, the length of a ruler. All of these have a clear starting point and ending point.
3.2. Rays
A ray is a part of a line that has one fixed starting point and extends endlessly in one direction. So a ray has a starting point but no end point.
If a ray starts at point \( A \) and passes through point \( B \), we write it as \( \overrightarrow{AB} \). The first letter in the symbol is always the starting point (the endpoint) of the ray.
Examples: a torch beam starting from the torch, a laser pointer light from its source, or a sunbeam entering a room from a small opening.
4. Naming and Notation of Points, Lines, Rays and Line Segments
To read and write geometrical diagrams easily, we follow some standard ways of naming objects:
- Point: written simply as \( A \), \( B \), \( P \) etc.
- Line: often written as line \( AB \) or using the symbol \( \overleftrightarrow{AB} \), meaning a line passing through points \( A \) and \( B \).
- Line segment: written as \( \overline{AB} \), which means the segment from point \( A \) to point \( B \).
- Ray: written as \( \overrightarrow{AB} \), which means a ray starting at point \( A \) and going through point \( B \) in that direction.
For rays, the order of letters is important because it shows which point is the starting point. For line segments, the order does not matter, so \( \overline{AB} \) and \( \overline{BA} \) represent the same segment.
5. Everyday Examples of Points, Lines, Rays and Line Segments
We see these basic geometrical ideas all around us:
- Points: the exact corner of a book, a dot made by a sharpened pencil, the location of a city on a map.
- Line segments: the sides of a notebook, the edges of a mobile phone, the handle of a door.
- Rays: a torch beam in a dark room, sunlight entering through a small window, light from a projector.
- Lines (ideal): a perfectly straight road imagined to extend forever, or railway tracks thought of as continuing without end in both directions.
Thinking of such real-life examples makes it easier to remember the meanings of points, lines, rays and line segments while solving geometry questions later.