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Algebra: How to Find the Length of a Line

Algebra Word Problem
Finding the length of a line when given the coordinates of the two end points.

Problem:

A line extends from the point (x=5, y=1) to the point (x=10, y=20). (See figure one)

What is the length of the line?

Solution:

For purposes of this discussion, I will call the line extending from (x=5, y=1) to (x=10, y=20) line c.

In order to solve this problem, we need to form a triangle with the given line being the hypotenuse of the triangle. Refer to figure two for this explanation.

The first thing we do is draw a vertical line from the point (x=10, y=20) to the point (x=10, y=1). As you can see in figure two, the line is parallel to the y axis. For purposes of this discussion, I will call this line line b.

Next, we draw a line from the point (x=5, y=1) to the point (x=10, y=1). As you can see in figure two, the line is parallel to the x axis. For purposes of this discussion, I will call this line line a.

By drawing these two lines, we have formed a right triangle.

The length of the vertical line can be found by subtracting like coordinates. The two coordinates are (x=10, y=20) and (x=10, y=1). The distance along the y axis is the difference 20 – 1 or 19. Line b is 19 units long.

The length of the horizonl line can be found by subtracting like coordinates. The two coordinates are (x=5, y=1) and (x=10, y=1). The distance along the y axis is the difference 10 – 5 or 5. Line a is 5 units long.

We now have the length of two sides of the triangle. The vertical line is 19 units long and the horizontal line is 5 units long.

Line a and line b intersect each other at an angle of 90 degrees. The triangle is a right trangle and therefore, we can apply pythagoream’s theorem to this problem.

According to Pythagoream’ theorem, the following equation can be used to find the length of line c.

(line c length)2 = (line a length)2 + (line b length)2 =

(19)2 + (5)2 =

361 + 25 =

386

(line c length)2 = 386

line c length = (386)1/2 = 19.64688270

Reference:
New Second Algebra
Library of Congress Catalog Card Number 62-7240

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