Lesson Explainer: Distance on the Coordinate Plane: Horizontal and Vertical Mathematics • 6th Grade

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‘₯,𝑦) and 𝐡(π‘₯,𝑦) can be found by applying the Pythagorean theorem. It gives 𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

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 𝑂𝐡+𝐴𝐡=𝑂𝐴3+4=𝑂𝐴.

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‘₯,𝑦), 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‘₯βˆ’π‘₯|. And we know that π‘₯=π‘₯, so we have 𝐢𝐡=|π‘₯βˆ’π‘₯|=10βˆ’7=3.

Similarly, the length of 𝐴𝐢 is 𝐴𝐢=|π‘¦βˆ’π‘¦|, and as 𝑦=π‘¦οŒ’οŒ‘, we find 𝐴𝐢=|π‘¦βˆ’π‘¦|=|9βˆ’5|=4.

Applying the Pythagorean theorem in △𝐴𝐡𝐢 gives 𝐴𝐡=𝐢𝐡+𝐴𝐢𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

𝐴𝐡 is a length, so we can write 𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)𝐴𝐡=√3+4=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‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).
  3. We plug the coordinates of 𝐴 and 𝐡 into the above equation: 𝐴𝐡=√(0βˆ’(βˆ’3))+(βˆ’3βˆ’4)𝐴𝐡=√3+(βˆ’7)𝐴𝐡=√9+49𝐴𝐡=√58.lengthunits

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‘₯,𝑦) or 𝐢(π‘₯,𝑦).

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‘₯βˆ’π‘₯+ο€Ήπ‘¦βˆ’π‘¦ο…οŒ²οŒ°οŠ¨οŒ²οŒ°οŠ¨.
  3. We plug the coordinates of 𝑄 and 𝑆 into the above equation: 𝑄𝑆=√(βˆ’1βˆ’5)+(3βˆ’6)𝑄𝑆=√(βˆ’6)+(βˆ’3)𝑄𝑆=√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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