Explainer: Distance on the Coordinate Plane: Horizontal and Vertical

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 𝐴(π‘₯𝐴,𝑦𝐴) and 𝐡(π‘₯𝐡,𝑦𝐡) can be found by applying the Pythagorean theorem. It gives 𝐴𝐡=(π‘₯π΅βˆ’π‘₯𝐴)2+(π‘¦π΅βˆ’π‘¦π΄)2.

Example 1: Finding the Distance from a Point to the Origin

Let us consider 𝐴(3,4), 𝐡(3,0), and 𝐢(0,4) in a coordinate system of origin 𝑂(0,0). Using the Pythagorean theorem, find the distance between 𝑂 and 𝐴.

Answer

We know from the given coordinates that 𝐴 and 𝐡 lie on a vertical line, as well as 𝑂 and 𝐢, as the two points in each pair have the same π‘₯-coordinate. Similarly, 𝐴 and 𝐢 lie on a horizontal line, as well as 𝑂 and 𝐡. 𝐴𝐡𝑂𝐢 is thus a rectangle, and △𝐴𝐡𝑂 and △𝐴𝐢𝑂 are congruent right triangles, with their right angles, respectively, at 𝐡 and 𝐢.

The distance between 𝑂 and 𝐴 is the length of 𝑂𝐴, and it can be found by applying the Pythagorean theorem either in △𝐴𝐡𝑂 or in △𝐴𝐢𝑂. In △𝐴𝐡𝑂, for instance, 𝑂𝐴 is the hypotenuse, while 𝑂𝐡 and 𝐡𝐴 are the legs. This gives 𝑂𝐡2+𝐴𝐡2=𝑂𝐴232+42=𝑂𝐴2.

We find that 𝑂𝐴=5.

Example 2: Finding the Distance between Two Points Using the Pythagorean Theorem

Let us now consider 𝐴(7,9) and 𝐡(10,5) in a coordinate system of origin 𝑂(0,0). △𝐴𝐡𝐢 has a right angle at 𝐢 and is such that 𝐡𝐢 is parallel to the π‘₯-axis and 𝐴𝐢 is parallel to the 𝑦-axis. Using the Pythagorean theorem, find the distance between 𝐴 and 𝐡.

Answer

We see that the coordinates of 𝐢 are completely defined by those of 𝐴 and 𝐡: they are (π‘₯𝐴,𝑦𝐡), here (7,5).

In △𝐴𝐡𝐢, the hypotenuse is 𝐴𝐡 and the legs are 𝐴𝐢 and 𝐢𝐡. Since 𝐢 and 𝐡 lie on a horizontal line, the length of 𝐢𝐡 is simply given by 𝐢𝐡=|π‘₯π΅βˆ’π‘₯𝐢|. And we know that π‘₯𝐢=π‘₯𝐴, so we have 𝐢𝐡=|π‘₯π΅βˆ’π‘₯𝐴|=10βˆ’7=3.

Similarly, the length of 𝐴𝐢 is 𝐴𝐢=|π‘¦πΆβˆ’π‘¦π΄|, and as 𝑦𝐢=𝑦𝐡, we find 𝐴𝐢=|π‘¦π΅βˆ’π‘¦π΄|=|9βˆ’5|=4.

Applying the Pythagorean theorem in △𝐴𝐡𝐢 gives 𝐴𝐡2=𝐢𝐡2+𝐴𝐢2𝐴𝐡2=(π‘₯π΅βˆ’π‘₯𝐴)2+(π‘¦π΅βˆ’π‘¦π΄)2.

𝐴𝐡 is a length, so we can write 𝐴𝐡=(π‘₯π΅βˆ’π‘₯𝐴)2+(π‘¦π΅βˆ’π‘¦π΄)2𝐴𝐡=√32+42=5.

The equation we have established to find the distance between points 𝐴 and 𝐡 is true for any couple of points.

Example 3: Finding Distances on a Coordinate Plane

Find the distance between the points 𝐴 and 𝐡.

Answer

  1. We find the coordinates of points 𝐴 and 𝐡 from the diagram: 𝐴(βˆ’3,4) and 𝐡(0,βˆ’3).
  2. We knowβˆ— that the distance between 𝐴 and 𝐡 is the length of 𝐴𝐡 and is 𝐴𝐡=(π‘₯π΅βˆ’π‘₯𝐴)2+(π‘¦π΅βˆ’π‘¦π΄)2.
  3. We plug the coordinates of 𝐴 and 𝐡 into the above equation: 𝐴𝐡=√(0βˆ’(βˆ’3))2+(βˆ’3βˆ’4)2𝐴𝐡=√32+(βˆ’7)2𝐴𝐡=√9+49𝐴𝐡=√58lengthunits.

βˆ—Note that if you do not remember the formula, it is very easy to retrieve it by applying the Pythagorean theorem in △𝐴𝐡𝐢 with 𝐢(π‘₯𝐴,𝑦𝐡) or 𝐢(π‘₯𝐡,𝑦𝐴).

Example 4: Finding Distances on a Coordinate Plane

Quadrilateral 𝑃𝑄𝑅𝑆 has vertices 𝑃(βˆ’1,6), 𝑄(5,6), 𝑅(5,3), and 𝑆(βˆ’1,3). Find the length of 𝑄𝑆.

Answer

  1. We find the coordinates of points 𝑄 and 𝑆 from the diagram: 𝑄(5,6) and 𝑆(βˆ’1,3).
  2. We know that the length of 𝑄𝑆 is 𝑄𝑆=π‘₯π‘†βˆ’π‘₯𝑄2+ο€Ήπ‘¦π‘†βˆ’π‘¦π‘„ο…2.
  3. We plug the coordinates of 𝑄 and 𝑆 into the above equation: 𝑄𝑆=√(βˆ’1βˆ’5)2+(3βˆ’6)2𝑄𝑆=√(βˆ’6)2+(βˆ’3)2𝑄𝑆=√36+9𝑄𝑆=√45,
    and as 45=9β‹…5,𝑄𝑆=√9β‹…5,
    and because βˆšπ‘Žβ‹…π‘=βˆšπ‘Žβ‹…βˆšπ‘,𝑄𝑆=√9β‹…βˆš5𝑄𝑆=3√5.

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