Line Segments - Revision Notes

In these revision notes for Line Segments, we cover the following key points:

Line Segments Revision Notes

Segments, intervals, half-segments and half-intervals can be shown on a number line. A filled (black) dot is a symbol used for segment (the endpoints are included in the set) while a blank (white) dot indicates an interval (the endpoints are not included in the set). Half-segments have the left endpoint shown by a filled dot and the right endpoint by a blank one, while half-intervals have the left endpoint shown by a blank dot and the right endpoint by a filled one.

A line segment is the part of a line that lies between two given endpoints. The difference between line segments and number sets is that line segments may lie in other directions besides the number axis because the orientation of a line is not always horizontal or vertical only. We usually refer to line segments when dealing with linear graphs. Line segments are useful when we want to focus our attention on a particular section of the graph.

Line intervals are like line segments but without including the two endpoints.

The line segments [AB] and [BA] express the same thing: both of them express the part of a line enclosed within the endpoints A and B.

The midpoint of a line segment is a point in that segment which has the same distance from both endpoints. In other words, the midpoint M of the segment AB is halfway of the straight path that brings from A to B or from B to A.

When trying to find the midpoint of a line segment AB, it is better to make it express in two different equations, according to the two main directions. We must add the coordinates of points A and B for each direction separately and divide by 2 the sums obtained. In this way, we obtain the x- and y-coordinates of the midpoint M. In symbols, we have

and

where A is the leftmost point of the two. We can also express the two endpoints by the indices (1) and (2) respectively, as subscripts. In this way, we obtain the general form of the two midpoint formulas above:

and

What is the importance of the midpoint of a segment? Well, there are several reasons why knowing the coordinates of the midpoint of a segment is a good thing, where the most important is to check whether a given graph is linear or not. For example, we may face a marginal part of a parabola produced by a quadratic graph that at the first sight looks linear but it isn't instead.

In most cases, line segments have a certain steepness, as well as the lines that contain them. Therefore, we must account for the change in coordinates in both directions when calculating the length of a line segment.

In general (unless the line is purely horizontal or vertical), a line segment and the two corresponding segments according to the two basic directions form a right triangle as the one shown in the figure below, where the x-and y-components of the segment AB, namely AB_x and AB_y, form the right angle, while the original segment AB forms the third side of the triangle, known in geometry as the hypotenuse. The length of a line segment is therefore calculated by applying the Pythagorean Theorem:

Basically, the procedure for calculating the length of a line segment is the same as that used for calculating the distance between two points with known coordinates. This is obvious, as the length of the shortest path from one endpoint to another corresponds to the length of the segment itself, as both are linear.

We can use ratios to identify and express the position of a given point in a line segment. The rules are the same as in other situations involving ratios. Obviously, the ratio rules are applied for each direction separately to find each of the coordinates of point P.

We can also find a missing endpoint of a line segment when the other endpoint and the coordinates of a point inside the segment - which is in a given ratio from the two endpoints - are given.