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