9.01 Geometrical diagrams
Introduction
Geometry is the study of shapes, and is one of the oldest areas of mathematical interest. Geometrical diagrams have many important features, and the terminology used in the subject is important to ensure good mathematical communication.
Points and segments
A point is a single location, with no height or width. We use capital letters to distinguish two different points.
If we connect all of the points from one point to another, we make a segment.
Exploration
Explore the applet to investigate segments. Make a selection and drag the coloured point to highlight all the points that lie on that segment:
A segment is a composition of points connected together between two endpoints.
We place small markings on segments when we want to show that they are equal in length.
This does not mean that the two segments are made up of the same points - only that they have the same length. Sometimes we will use more than one kind of marking to show that some segments are equal to others.
Examples
Example 1
This diagram has two equal segments marked:
What segment is equal in length to ZY?
A point is a single location, with no height or width and can be labelled as a single letter like A or B.
A segment is a composition of points connected together between two endpoints.
Rays
If we start at one point and keep going, we make a ray.
Direction is important for rays - these two objects are not the same.
Exploration
Explore the applet to investigate rays. Make a selection and drag the coloured point to highlight all the points that lie on that ray:
The ray extends through the second point in the name of ray. For instance, for Ray {AB}, the ray extends through B, not A.
A ray is a composition of points connecting two points that goes through one of those points.
The ray extends through the second point in the name of ray. For instance, for Ray {AB}, the ray extends through B, not A.
Lines
If we keep going in both directions, we make the line through A and B:
Exploration
Explore the applet to investigate lines. Make a selection and drag the coloured point to highlight all the points that lie on that line:
A line is a composition of points that passes through two points.
To summarise, segments and lines stay the same if we reverse the order of the points, but this is not true for rays:
Examples
Example 2
Select the diagram that shows the line through E and F:
A line between two points contains every point between the points and all the points beyond on either side. A ray starts at one point and continues through another and beyond. A segment starts at one point and stops at the other.
Rays go through different directions when the points are reversed which makes it different from segments and lines.
Angles
Whenever two lines, rays, or segments pass through the same point, we can describe the relative orientation of one to the other using an angle.
We can use three points to refer to an angle by using the symbol “\angle" followed by three letters, one for each point. The first letter will be on one of the rays, lines, or segments, the second point will be their intersection, and the third will be a point on the other ray, line, or segment. This means there are two equally valid ways to refer to an angle, as illustrated in this diagram:
Just like with segments, we can use additional markings to show that two angles are equal. We draw multiple arcs to show that different angles are equal to each other.
Examples
Example 3
This diagram has the angle \angle {ABC} marked. What is another way of referring to the same angle?
The angle between two intersecting segments, lines, or rays represents their relative orientation to each other. We write the angle symbol “\angle" followed by three letters. The second letter is always the intersection point, and the first and third letters lie on the objects forming the angle, one on each.
Swapping the first and third letters does not change the angle.