University:
College
Course:
College Mathematics
Academic year:

2023
Views:

438

Pages:

2
Author:

anon

Mathematics: Distance between two coordinates We want to use coordinates in more sophisticated ways now that we understand how they function. This lesson we are going to learn how we can calculate the distance between two coordinates. However, to make the most out of this lesson, I suggest you first have a solid understanding of the Pythagoras Theorem. Ok, let’s first do a simple warm-up. I have a Cartesian Plane below, and there are two coordinates plotted below as well. As a recap to the previous lesson, let’s find these two coordinates, 1 and 2. Coordinate 1: x=2, y=5, so the coordinate, in the form (x,y), is (2,5) Coordinate 2: x=8, y=2, so the coordinate, in the form (x,y), is (8,5) Now, these two problems was just a starter for what is going to be a more difficult problem in a second. We want to find the distance between the two points. Now, this isn’t a very difficult problem. If you connect the two dots, you will notice that the answer is quite straightforward and can be solved by anyone who has taken a basic subtraction course and knows the Cartesian plane relatively well. The distance between Coordinates 1 and 2 is simply the difference between the x-coordinate values as the y-values are in the same plane. This question is as easy as asking, Joe is standing in the coffee shop, which we can call…x=2, whilst Maggie is standing directly to the right, in the Police Station, which we call x=8. The distance between Joe and Maggie is then simply 8-2 = 6, which is the answer to the problem here. But, what if Maggie is in the hospital but the hospital is not directly to the right or left…What if the hospital is North east to the coffee shop Joe is in? Look back at the Cartesian and I will introduce you to coordinates 3 and 4. As usual, let’s find the coordinates of Coordinates 3 and 4. In Coordinate 3, x=5, y=4, so the coordinate is (5,4). In Coordinate 4, x=2, y=1, so the coordinate is (2,1) How can we find the distance? Let’s first connect the dots. If you connect the dots of Coordinates 3 and 4, you will notice that I just made a right angle triangle and the distance between coordinates 3 and 4 is the hypotenuse! If you can find the distance between the other two sides of the triangle, the adjacent and opposite sides, you can then subsequently use the Pythagoras Theorem to calculate the length of the hypotenuse. I labeled one side, Side A and the other side, Side B. I will label the hypotenuse, the side we want to find, Side C. Definition to learn: “if the distance between two points has one mutual x or y coordinate, the distance between the two coordinates is simply the direct distance between the two non-equal coordinates. E.g. the distance between (2,2) and (4,2) will be the distance between 2 and 4, which is 2, as the y-coordinates here are mutual” Length of side A = (2,1) and (5,1) are the two coordinates. Using the definition above, the length of side A is 3. Length of side B= (5,1) and (5,4) are the two coordinates. Again, using the definition above, the length of side B is 3. Hence, using Pythagoras’s Theorem: As (Side C)2 = (Side A)2 + (Side B)2 (Side C)2 = 32+32 (Side C)2 = 18 Side C = √18 Formula for distance = √(y-y2)2 +(x-x2)2