Math Lesson 14.5.4 - Calculating the Length of a Line Segment

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Welcome to our Math lesson on Calculating the Length of a Line Segment, this is the fourth lesson of our suite of math lessons covering the topic of Line Segments, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Calculating the Length of a Line Segment

As mentioned earlier in this tutorial, line segments are not always horizontal or vertical only. 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.

There is a theorem in geometry, known as the Pythagoras Theorem, according to which, we add the squares of the two sides that form the right angle and this sum corresponds to the square of the hypotenuse. (A theorem in geometry is a statement that needs proof to be confirmed as true). We will dedicate an entire chapter to Pythagorean Geometry, but for now, it is sufficient to give the formula derived from the Pythagorean Theorem. In our case, this theorem is written as

In this way, if we know the coordinates of points A and B, we find the length of the given line segment AB by calculating the values of AB_x and AB_y first, then we raise these component segments in the second power and add them. This gives the square of the length of the segment AB. Last, we find the square root of the above number to find the length of the line segment AB.

Basically, the procedure for calculating the length of a line segment is the same as that used in the previous tutorial 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.

Remarks!

1. You will see in some textbooks the line segment AB denoted as (A̅B̅) (a horizontal line above AB) instead of [AB] or in words, i.e. the "segment AB". All these notations represent the same thing.
2. AB_x and AB_y described above are nothing more than ΔAB_x and ΔAB_y we have seen in other cases, which are obtained by subtracting the coordinates of B and A according to the given direction. In this way, we obtain the following equivalent expressions.
   AB_x = ∆AB_x = x_B - x_A
   and
   AB_x = ∆AB_x = y_B - y_A
   where A is the leftmost point of the two.

Example 4

Calculate the perimeter (the sum of all sides) of the triangle ABC shown in the figure below. Write the answer to one decimal place.

Solution 4

First, we identify the coordinates of points A, B and C in the figure. Thus, point A is 8 units on the left and 2 units above the origin. Hence, we have A(-8, 2). Likewise, point B is 1 unit on the right and 4 units below the origin. Hence, we have B(1, -4). Last, point C is 5 units on the right and 9 units above the origin. Hence, we have C(5, 9).

Next, we calculate the line segments AB, AC and BC according to the procedure described above in theory. Thus, for the segment AB we have

and

Thus, the length of the line segment AB raised in square is

Hence, the length of AB is

The same procedure is followed for the line segment AC as well. Thus,

and

Thus, the length of the line segment AC raised in square is

Hence, the length of AC is

Last, for the line segment BC, we have

and

Thus, the length of the line segment BC raised in square is

Hence, the length of BC is

Now, let's calculate the perimeter P of triangle ABC by adding the three line segments found above. We have

More Line Segments Lessons and Learning Resources

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Enjoy the "Calculating the Length of a Line Segment" math lesson? People who liked the "Line Segments lesson found the following resources useful:

1. Length Feedback. Helps other - Leave a rating for this length (see below)
2. Linear Graphs Math tutorial: Line Segments. Read the Line Segments math tutorial and build your math knowledge of Linear Graphs
3. Linear Graphs Revision Notes: Line Segments. Print the notes so you can revise the key points covered in the math tutorial for Line Segments
4. Linear Graphs Practice Questions: Line Segments. Test and improve your knowledge of Line Segments with example questins and answers
5. Check your calculations for Linear Graphs questions with our excellent Linear Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Linear Graphs Calculators by iCalculator™ below.
6. Continuing learning linear graphs - read our next math tutorial: Linear Graphs

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