Distance Between Points

The distance between two points in mathematics is a measure of how far apart those two points are in space. It’s a fundamental concept in geometry, and it’s used in various fields of mathematics and science. The distance between two points can be calculated in different ways, depending on whether you are working in a one-dimensional, two-dimensional, or three-dimensional space.

Here, I’ll explain how to calculate the distance between points in two and three dimensions, which are the most common cases.

Distance between Two Points in Two-Dimensional Space

In a two-dimensional (2D) plane, such as a piece of paper, the distance between two points (x1, y1) and (x2, y2) can be calculated using the Pythagorean theorem. The formula is:

Distance = √((x2 – x1)² + (y2 – y1)²)

Here’s how it works:

(x1, y1) represents the coordinates of the first point.

(x2, y2) represents the coordinates of the second point.

(x2 – x1) represents the horizontal (x) difference between the two points.

(y2 – y1) represents the vertical (y) difference between the two points.

The square of these differences is added together, and then the square root is taken to get the distance.

The following explain makes you understand better by means of an example given below:

Solved Problems

Problem 1

Find the distance between the points A(3, 4) and B(6, 8). 

Solution

To find the distance between two points (x₁, y₁) and (x₂, y₂), you can use the distance formula:

Distance = √((x₂ – x₁)² + (y₂ – y₁)²)

In this case:

Distance = √((6 – 3)² + (8 – 4)²)

Distance = √(3² + 4²)

Distance = √(9 + 16)

Distance = √25

Distance = 5 units

So, the distance between A(3, 4) and B(6, 8) is 5 units.

Problem 2

Calculate the distance between the points C(-2, 5) and D(1, -1).

Solution

Using the distance formula again:

Distance = √((1 – (-2))² + (-1 – 5)²)

Distance = √(3² + (-6)²)

Distance = √(9 + 36)

Distance = √45 units

Problem 3

Find the distance between points E(0, 0) and F(3, 4).

Solution

Using the distance formula:

Distance = √((3 – 0)² + (4 – 0)²)

Distance = √(3² + 4²)

Distance = √(9 + 16)

Distance = √25

Distance = 5 units

Problem 4

Determine the distance between points G(5, -2) and H(-3, 7). 

Solution

Using the distance formula:

Distance = √((-3 – 5)² + (7 – (-2))²)

Distance = √((-8)² + (7 + 2)²)

Distance = √(64 + 81)

Distance = √145 units

Problem 5

What is the distance between points I(-1, -1) and J(4, 5)? 

Solution

Using the distance formula:

Distance = √((4 – (-1))² + (5 – (-1))²)

Distance = √(5² + 6²)

Distance = √(25 + 36)

Distance = √61 units

Problem 6

Calculate the distance between points K(2, 3) and L(2, -2).

Solution

Using the distance formula:

Distance = √((2 – 2)² + (-2 – 3)²)

Distance = √(0² + (-5)²)

Distance = √25

Distance = 5 units

Problem 7

Find the distance between points M(1, 1) and N(7, 7).

Solution

Using the distance formula:

Distance = √((7 – 1)² + (7 – 1)²)

Distance = √(6² + 6²)

Distance = √(36 + 36)

Distance = √72 = 6√2 units

Problem 8

Determine the distance between points O(-3, 2) and P(5, 2).

Solution

Using the distance formula:

Distance = √((5 – (-3))² + (2 – 2)²)

Distance = √(8² + 0²)

Distance = √64

Distance = 8 units

Problem 9

Calculate the distance between points Q(4, 1) and R(4, -5). 

Solution

Using the distance formula:

Distance = √((4 – 4)² + (-5 – 1)²)

Distance = √(0² + (-6)²)

Distance = √36

Distance = 6 units

Problem 10

What is the distance between points S(0, 0) and T(-3, -4)?

Solution

Using the distance formula:

Distance = √((-3 – 0)² + (-4 – 0)²)

Distance = √((-3)² + (-4)²)

Distance = √(9 + 16)

Distance = √25

Distance = 5 units

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