Distance Between Two Points

The term “distance” is a numerical measurement to determine how far two points or objects are in a two-dimensional space. In Geometry, the distance is always positive, but not when the points coincide. The distance from point X to Y = distance from point Y to X. The distance from point X to Y is represented as |XY|.

The distance between two points is the length of the path that joins the two given points. We can use the distance formula to determine the distance between two points. To derive the formula for calculating the distance between two points, we can consider two points A and B.

We can construct a right-angled triangle ABC, as shown in the figure ²given below. In the given right-angled triangle ABC, point A has coordinates (a,b), Point B has coordinates (c,d), and point C has coordinates (a,d).

Now, we can apply the Pythagoras theorem to determine the distance between two points i.e. A and B.

AB = |b – d|2 + |a – c|2

The above equation can also be expressed as (a – c)2 + (b – d)2


Distance Formula

The distance formula is a mathematical formula that is used to determine the distance between points. The two points can be in any dimension.

For example, you can find the distance between two points on a line (one-dimensional), on a plane (two-dimensional), or on a space (three-dimensional). This is an important topic and can be learned in an interactive way from Cuemath.


Distance Formula in One Dimension

Let P = (x) and Q = (x) are two different points that fall on a real number line.

Then the distance between P and Q is calculated as

d P,Q

In the plane, if we consider the x-axis as a one-dimensional line, then we can find the distance between two points falls on the x-axis as the absolute value of the difference of their x-coordinates.

Similarly,  if we consider the y-axis as a one-dimensional line, then we can find the distance between two points falls on the y-axis as the absolute value of the difference of their y-coordinates.


Distance Formula In Two Dimension.

To derive the distance formula in two dimensions, consider the XY- plane, and let P and Q be two points on it. The coordinates of P are given as (x, y) whereas the coordinates of Q are given as (x, y). Therefore, the distance between two points P and Q with given coordinates is calculated using the Pythagorean theorem as:

PQ


Distance Formula Examples

Example 1: Calculate the distance between the points (0,4) and (0,12).

Solution:

As we can see both of the points lie on the y-axis and hence the distance between the given two points on the y-axis is the absolute value of the difference of y-coordinates.

Accordingly:

d

Therefore, the distance between the points (0,4) and (0,12) is 8 units.

Example 2: Find the distance between the points (4,4) and (7,8).

Solution:

Here,

x

The given problem can be simply solved by substituting the values in the below-given distance formula:

d =

Therefore, the distance between the two points (4,4) and (7,8) is 5 units.

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