Points, Lines and Planes
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Written Lesson
Geometry begins with simple objects such as points, lines, line segments, rays, and planes. These are the basic building blocks used to describe shapes, angles, and diagrams in CSEC Mathematics.
What You Will Learn
- Name the basic geometry terms: point, line, line segment, ray, and plane
- Use correct notation to name and write about these figures
- Describe the difference between a line, line segment, and ray
- Identify collinear and coplanar points
- Identify parallel, perpendicular, and intersecting lines
- Describe how planes relate to each other
- Recognize points, lines, and planes in everyday objects and solid shapes
Key Ideas
- A point shows an exact position. It has no length, width, or size.
- A line is straight and continues forever in both directions.
- A line segment is part of a line with two endpoints.
- A ray has one endpoint and continues forever in one direction.
- A plane is a flat surface that extends in all directions.
Point, Line, Line Segment, and Ray
These four terms are often used in geometry diagrams. It is important to know how they are different.
Naming These Figures
Mathematicians use short symbols to name points, lines, line segments, and rays without drawing them each time.
| Figure | How it is named | What it means |
|---|---|---|
| Point | A | A single capital letter |
| Line | ↔AB | Passes through A and B, goes on forever both ways |
| Line segment | ¯AB | Stops at endpoints A and B |
| Ray | →AB | Starts at A, passes through B, continues forever past B |
Line Segment
A line segment has two endpoints. It does not continue forever. For example, the line segment below starts at A and ends at B.
Ray
A ray has one endpoint and goes on forever in one direction.
Comparing Lines, Line Segments, and Rays
| Figure | Number of Endpoints | Description |
|---|---|---|
| Line | 0 | Continues endlessly in both directions |
| Line segment | 2 | Stops at both endpoints |
| Ray | 1 | Continues endlessly in one direction |
Collinear Points
Points that lie on the same straight line are called collinear points.
Example: Are points A, B and D collinear?
Answer: No. The three points do not lie on the same straight line, so A, B and D are not collinear.
Plane
A plane is a flat surface. It extends in all directions. A page, a tabletop, or a wall can help you imagine a plane.
A plane can be named using three points inside it, as long as the three points are not collinear. The plane shown above can be called plane ABC.
Coplanar Points
Points that lie in the same plane are called coplanar points. Points that do not all lie in the same plane are called non-coplanar points.
How Lines Can Relate to Each Other
- Parallel lines are always the same distance apart and never meet.
- Perpendicular lines meet at a right angle, which is 90°.
- Intersecting lines cross or meet at a point.
Parallel Lines
Parallel lines never meet, even if they are extended. The symbol for "is parallel to" is ∥. If line AB is parallel to line CD, we write AB ∥ CD.
Perpendicular Lines
Perpendicular lines meet at a right angle. A right angle is 90°. The symbol for "is perpendicular to" is ⊥. If line AB is perpendicular to line CD, we write AB ⊥ CD.
Intersecting Lines
Intersecting lines meet or cross at one point. They do not have to meet at 90°.
How Planes Can Relate to Each Other
Planes can also intersect or stay parallel, in the same way lines can.
- Intersecting planes meet along a straight line. For example, a wall and the floor of a room are two intersecting planes, and the edge where they meet represents a line.
- Parallel planes never meet. For example, the floor and ceiling of a room, the opposite faces of a rectangular box, and two level shelves placed one above the other are all examples of parallel planes.
Points, Lines, and Planes in Solid Shapes
The ideas from this lesson also describe the parts of solid shapes. Consider a rectangular prism, such as a box.
- Each corner represents a point.
- Each edge represents a line segment.
- Each flat face represents part of a plane.
A rectangular prism has:
- 8 vertices, or points
- 12 edges, or line segments
- 6 faces, or parts of planes
Worked Examples
Example 1: Two straight roads cross each other so that the angle between them is 90°. How are the roads related?
- The roads meet at a single point, so they intersect.
- The angle between them is exactly 90°.
- Lines that meet at 90° are called perpendicular lines.
Answer: The roads are perpendicular.
Example 2: A drawing shows two straight railway tracks that stay the same distance apart and never meet. What type of lines are they?
- The tracks do not cross.
- They stay the same distance apart.
- Lines like this are parallel.
Answer: The railway tracks are parallel lines.
Example 3: A diagram shows a straight path from point A to point B. The path stops at A and B. Is this a line, line segment, or ray?
- The path has two endpoints.
- It does not continue forever.
- A straight path with two endpoints is a line segment.
Answer: It is a line segment, written ¯AB.
Example 4: A beam of light starts at point P and continues in one direction. Is this best described as a line, line segment, or ray?
- The beam starts at one point.
- It continues in one direction.
- A figure with one endpoint that continues in one direction is a ray.
Answer: It is a ray.
Example 5: A wall meets the floor of a room. What does their intersection represent?
- The wall and the floor represent two planes.
- Two intersecting planes meet along a straight line.
Answer: The intersection is a line.
Common Mistakes
Important Reminder
Lesson Summary
- A point represents an exact position and has no size.
- A line extends endlessly in both directions, written ↔AB.
- A line segment has two endpoints and a fixed length, written ¯AB.
- A ray has one endpoint and continues endlessly in one direction, written →AB, with the endpoint written first.
- Collinear points lie on the same straight line.
- A plane is a flat surface that extends endlessly, and can be named using three non-collinear points inside it.
- Coplanar points lie in the same plane.
- Parallel lines (∥) never meet. Perpendicular lines (⊥) meet at 90°. Intersecting lines meet at one point.
- Two intersecting planes meet along a line. Parallel planes never meet.
Practice Questions
- What is the difference between a line and a line segment?
- How many endpoints does a ray have?
- Two lines never meet, no matter how far they are extended. What are they called?
- Two lines meet at 90°. What are they called?
- What do we call the point where two lines cross?
- What geometric term describes an exact location with no size?
- Points P, Q and R lie on the same straight line. What word describes these points?
- Points A, B, C and D lie in the same plane. What word describes these points?
- Write the mathematical symbol meaning "is parallel to."
- What is formed when two planes intersect?
Answers
- A line continues endlessly in both directions, while a line segment has two endpoints and a fixed length.
- One endpoint
- Parallel lines
- Perpendicular lines
- The point of intersection
- A point
- Collinear points
- Coplanar points
- ∥
- A line
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