The image below shows what we mean by the distance between the points at (x1, y1) and (x2, y2):
x1, y1, x2 and y2 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: (x2 − x1)2 = (x2 − x1) × (x2 − x1)
and 2 means squared: (y2 − y1)2 = (y2 − y1) × (y2 − y1)
2
Find x1, y1, x2 and y2 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 (x1, y1) and (x2, y2).
(x1, y1) = (1, 1) ∴ x1 = 1, y1 = 1
(x2, y2) = (5, 4) ∴ x2 = 5, y2 = 4
3
Substitute x1, y1, x2 and y2 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:
AB2 = BC2 + CA2
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 = x2 − x1
- BC = y2 − y1
If we substitute this into Pythagoras' formula:
AB2 = BC2 + CA2
AB2 = (x2 − x1)2 + (y2 − y1)2
Finally, take the square root of both sides:
AB = √((x2 − x1)2 + (y2 − y1)2)
This page was written by Stephen Clarke.
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