The Lesson
The distance between two points with Cartesian coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}) can be found using the formula:The image below shows what we mean by the distance between the points at (x_{1}, y_{1}) and (x_{2}, y_{2}):
x_{1}, y_{1}, x_{2} and y_{2} are symbols that represent the x-coordinates and y-coordinates of the points. In real questions, the Cartesian coordinates will have numbers, for example (1, 1) and (5, 4).
How to Find the Distance Between Two Points
Finding the distance between two points is easy.Question
Find the distance between the points with Cartesian coordinates (1, 1) and (5, 4).Step-by-Step:
1
Start with the formula:
$$Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Don't forget: √ means square root
and ^{2} means squared: (x_{2} − x_{1})^{2} = (x_{2} − x_{1}) × (x_{2} − x_{1})
and ^{2} means squared: (y_{2} − y_{1})^{2} = (y_{2} − y_{1}) × (y_{2} − y_{1})
2
Find x_{1}, y_{1}, x_{2} and y_{2} from the Cartesian coordinates given in the question.
In our example, the Cartesian coordinates of the points are (1, 1) and (5, 4). They are represented in the formula by (x_{1}, y_{1}) and (x_{2}, y_{2}).
(x_{1}, y_{1}) = (1, 1) ∴ x_{1} = 1, y_{1} = 1
(x_{2}, y_{2}) = (5, 4) ∴ x_{2} = 5, y_{2} = 4
3
Substitute x_{1}, y_{1}, x_{2} and y_{2} into the formula.
$$Distance = \sqrt{(5 - 1)^2 + (4 - 1)^2}$$
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{4^2 + 3^2}$$
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{(4 \times 4) + (3 \times 3)}$$
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{16 + 9}$$
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{25}$$
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 5$$
Answer:
The distance between the points with Cartesian coordinates (1, 1) and (5, 4) is 5.Why Does the Formula Work?
The formula to find the distance between points is derived from Pythagoras' theorem. Imagine joining two points A and B with a line. A right triangle can be formed from this by drawing straight down and straight across from the points, meeting at C.Pythagoras' theorem tells us that the length of the diagonal line squared is equal to the sum of the squares of the length of the blue lines: AB^{2} = BC^{2} + CA^{2} As AB is the distance between the points, we need to know the lengths of the blue lines, BC and CA.
- CA is the horizontal distance between the points, which is given by the difference between their x-coordinates.
- BC is the vertical distance between the points, which is given by the difference between their y-coordinates.
- CA = x_{2} − x_{1}
- BC = y_{2} − y_{1}