### Theory:

Let us recall the distance formula learnt in the previous topics.
Distance formula is used to compute the distance between two definite points.
Distance formula for a line segment parallel to an axis:
Line segment parallel to $$y$$-axis: Let the points be $$A$$ and $$B$$, and let the coordinates be ($$x_1$$, $$y$$) and ($$x_2$$, $$y$$) respectively.

In both the coordinates, $$y$$ is common as both the points lie either on the same axis or parallel to the same axis.

In this case, the points lie either on the $$y$$-axis or parallel to $$y$$-axis

In such a case, the distance formula will be:

$$\text{Distance}$$ $$=$$ $$|x_1 - x_2|$$

Line segment parallel to $$x$$-axis: Let the points be $$A$$ and $$B$$, and let the coordinates be ($$x$$, $$y_1$$) and ($$x$$, $$y_2$$) respectively.

In both the coordinates, $$x$$ is common as both the points lie on the same axis or parallel to the same axis.

In this case, the points lie either on the $$x$$-axis or parallel to $$x$$-axis.

In such a case, the distance formula will be:

$$\text{Distance} = |y_1 - y_2|$$

Let us look at the following example.

Find the distance between the points given in the figure below.

The coordinates of point $$A$$($$x_1$$, $$y_1$$) is ($$2$$, $$3$$).

The coordinates of point $$B$$($$x_2$$, $$y_2$$) is ($$2$$, $$-2$$).

$$x_1 = x_2 = 2$$

$$y_1 = 3$$

$$y_2 = -2$$

Distance between the points $$A$$ and $$B$$ can be obtained using the distance formula.

From the coordinates given above, $$x$$-coordinate is the same across both the points.

Therefore, $$\text{Distance} = |y_1 - y_2|$$

$$= |2 - (-2)|$$

$$= |2 + 2|$$

$$= 4$$

Distance formula if the line segment is not parallel to an axis:

Not all line segments are parallel to an axis.

Let the points of the line segment be $$A$$ and $$B$$, and let the co-ordinates be ($$x_1$$, $$y_1$$) and ($$x_2$$, $$y_2$$) respectively.

In that case, the distance formula will be:

$$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Let us look at the following example.

Find the distance between the points given in the figure below.

The coordinates of point $$A$$($$x_1$$, $$y_1$$) is ($$6$$, $$4$$).

The coordinates of point $$B$$($$x_2$$, $$y_2$$) is ($$1$$, $$-2$$).

$$x_1 = 6$$

$$x_2 = 1$$

$$y_1 = 4$$

$$y_2 = -2$$

Distance between the points $$A$$ and $$B$$ can be obtained using the distance formula.

$$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

$$= \sqrt{(1 - 6)^2 + (-2 - 4)^2}$$

$$= \sqrt{(-5)^2 + (-6)^2}$$

$$= \sqrt{25 + 36}$$

$$= \sqrt{61}$$