How to Find the Distance Between Two Points (Mathematics Lesson)
Finding the Distance Between Two Points
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.An Example Question
Find the distance between the points with Cartesian coordinates (1, 1) and (5, 4).Step 1
Distance = √((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2})
Don't forget: √ means square rootand ^{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})
Step 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.
(x_{2}, y_{2}) = (5, 4) ∴ x_{2} = 5, y_{2} = 4.
Step 3
Substitute x_{1}, y_{1}, x_{2} and y_{2} into the formula.
Distance = √((5 - 1)^{2} + (4 - 1)^{2})
Distance = √((4)^{2} + (3)^{2})
Distance = √((4 × 4) + (3 × 3))
Distance = √(16 + 9)
Distance = √25
Distance = 5
The distance between the points with Cartesian coordinates (1, 1) and (5, 4) is 5.Distance = √((4)^{2} + (3)^{2})
Distance = √((4 × 4) + (3 × 3))
Distance = √(16 + 9)
Distance = √25
Distance = 5
A Real Example of How to Find the Distance Between Two Points
The slider below shows another real example of how to find the distance between two points:Interactive Test
showHere's a second test on finding the distance between points.
Here's a third test on finding the distance between points.
Note
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}
If we substitute this into Pythagoras' formula:
AB^{2} = BC^{2} + CA^{2}
AB^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}
Finally, take the square root of both sides:
AB = √((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2})