Subject: Geometry

Points Lines Planes

The Breakdown

Point

A symbol used to show position. Represented by a small dot or a letter.

Lets look at points. A dictionary definition with regard to geometry is ‘that which has position but not magnitude’. Basically this means that a point has no size, length, width, height, mass, weight or shape. What it does have is position.

A point is a marker of significance or importance within a diagram – the author is calling your attention to a certain place.

For the purposes of most geometrical thought, at the exact position where two or more lines meet, like the corner of a square or triangle, there is a point, although these are not always marked with a dot. They may be if we are being asked a question in a test, however, or simply to call attention to that position.

It is important that when we are working with shapes that we clearly label any important points. Sometimes we’ll only need to label one; sometimes we’ll need to label all of them. To save time and to avoid confusion you need only label the important points of any diagram you are drawing (obviously unless you’re told otherwise).

For example, if we’re a builder, architect, electrician or another tradesperson we’ll need to be precise in communicating which wall we would like knocked down, or where we want the light switches. Whether we’re demonstrating our knowledge in a test or assignment, or drawing a diagram for a home renovation, we need to be very clear in our labelling of points.

We use points to show position, a place of importance. They have no size. A point is denoted by a dot. While dots may be quite large on paper, they mark only one position, not an area. A dot has size while a point has not. The place where two or more lines meet is a point.

Lines

A line has no width or area, but it does have length.

The word ‘line’ in geometry usually means a line of infinite length, or potentially infinite length, unless otherwise mentioned. Often geometers will put arrows on either end of a line to show it continues infinitely.

When using real world examples we are not saying width and area do not exist, just that for the purposes of one equation they do not matter, their importance is negligible. In geometry, the term line usually means a straight line. However, arcs are still considered lines, even though they are circular.

While a line may be represented by something like a physical rope or a thick marker on a whiteboard, it does not have measureable thickness. If it does, it becomes a rectangle or a cuboid.

The shortest distance between two points is always a straight line, or to be precise, a segment. A straight line segment is the portion of a straight line between and including two (or sometimes more) points. The expression straight line segment can be shortened to line segment, or simply to segment.

Lines are often labelled with letters at their points. This is the line segment AB.

Points Lines and Planes
Figure 1: Diagram of line AB

We write this \bar{AB} , with an overbar – a line above the two letters. This is a formal naming technique in geometry, it basically means “the finite line segment between point A and point B”. You’ll also see people write it AB, without the overbar, which means the same thing, it’s just not as formal, and it saves time. It’s important to label, so that we may make statements about their length, and other spatial relationships they may or may not have with other shapes.

For instance, we might say that AB is greater than CD, or AB > CD:

Points Lines and Planes
Figure 2: LINE CD, SHORTER THAN AB

Rays

A ray is a portion of a straight line that extends limitlessly from a given point. To designate rays, geometers will put an arrow on only one end of a segment. This is ray AB, or \vec{AB}. Note that the overbar has an arrow on one end.

Points Lines and Planes
Figure 3: Diagram of Ray AB

Bisectors

A line that crosses another line at exactly halfway between two points is said to bisect it. These lines cross at a right angle, therefore they are perpendicular bisectors with respect to each other.

Points Lines and Planes
Figure 4: Diagram of Bisector CD bisecting line AB

AX = BX is one way to demonstrate that point X, where line CD cuts line AB, is exactly halfway between points A and B. If the point of intersection was not halfway between A and B, we could demonstrate this by writing AX ≠ BX, or AX < or > BX.

Points Lines and Planes
Figure 5: Two lines cutting at perpendicular angles with an X at the point of obvious non-medial intersection

A set of points is said to be collinear (also spelt co-linear and colinear) if they lie upon a line. Points H, I, J and K are collinear, but not with point Z.

Points Lines and Planes
Figure 5: COLLINEAR POINTS

Planes

A plane is a flat, two-dimensional surface that has length and width, but no depth. A plane can be represented by a tv screen, a patch of road, a picture, or something like that. One side of a box could represent a plane, but the entire box could not. The surface of a sphere is a plane. Plane geometry is the study of plane figures; those that can be drawn on a plane.

A baseball field is an example of a plane.
Figure 5: A baseball field is an example of a plane