In this explainer, we will learn how to find the horizontal or vertical distance between two points on the coordinate plane.
Let’s look at a couple of examples to understand how it works.
Distance between Two Points in a Coordinate Plane
The distance between any two points and can be found by applying the Pythagorean theorem. It gives
Example 1: Finding the Distance from a Point to the Origin
Let us consider , , and in a coordinate system of origin . Using the Pythagorean theorem, find the distance between and .
Answer
We know from the given coordinates that and lie on a vertical line, as well as and , as the two points in each pair have the same -coordinate. Similarly, and lie on a horizontal line, as well as and . is thus a rectangle, and and are congruent right triangles, with their right angles, respectively, at and .
The distance between and is the length of , and it can be found by applying the Pythagorean theorem either in or in . In , for instance, is the hypotenuse, while and are the legs. This gives
We find that .
Example 2: Finding the Distance between Two Points Using the Pythagorean Theorem
Let us now consider and in a coordinate system of origin . has a right angle at and is such that is parallel to the -axis and is parallel to the -axis. Using the Pythagorean theorem, find the distance between and .
Answer
We see that the coordinates of are completely defined by those of and : they are , here .
In , the hypotenuse is and the legs are and . Since and lie on a horizontal line, the length of is simply given by . And we know that , so we have .
Similarly, the length of is , and as , we find .
Applying the Pythagorean theorem in gives
is a length, so we can write
The equation we have established to find the distance between points and is true for any couple of points.
Example 3: Finding Distances on a Coordinate Plane
Find the distance between the points and .
Answer
- We find the coordinates of points and from the diagram: and .
- We know that the distance between and is the length of and is .
- We plug the coordinates of and into the above equation:
Note that if you do not remember the formula, it is very easy to retrieve it by applying the Pythagorean theorem in with or .
Example 4: Finding Distances on a Coordinate Plane
Quadrilateral has vertices , , , and . Find the length of .
Answer
- We find the coordinates of points and from the diagram: and .
- We know that the length of is .
- We plug the coordinates of and into the above equation:
and as
and because
