The study of geometry is probably old as our civilization as the word 'geometry' means 'the measurement of the earth'.** Analytical Geometry**,
this title indicates that it deals with geometry by analytical methods.
Another name of analytic geometry is coordinate geometry, or Cartesian
geometry. Coordinate geometry is the study of geometry using a coordinate system to manipulate equations for straight lines,
planes etc. Analytic geometry is widely used in many fields of physics
and engineering.

In analytic geometry, the most common coordinate system
to use is the Cartesian coordinate system, where x-coordinates are
represents on horizontal axis, and y-coordinates are on vertical
axis. This system can also be used for 3D geometry.

In analytic geometry, distance between two points and angle measure are defined using formulas.

The distance between two points A(x$_1$, y$_1$) and B(x$_2$, y$_2$) is defined by the formula

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

and Slope of line (m) = tan $\theta$ = $\frac{y_2 - y_1}{x_2 - x_1}$

Let us consider some examples:

**Example 1**: Find the distance between the points (6, 9) and (8, 7).

**Solution:** Let 'd' be the distance between points (6, 9) and (8, 7)

We know that, formula for the distance between two points A(x$_1$, y$_1$) and B(x$_2$, y$_2$) is

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

Here, x$_1$ = 6, x$_2$ = 8, y$_1$ = 9 and y$_2$ = 7

d = $\sqrt{(8 - 6)^2 + (7 - 9)^2}$

= $\sqrt{4 + 4}$

= 2$\sqrt{2}$

Thus the distance between given points is 2$\sqrt{2}$.

**Example 2:** Determine the gradient of points (2, 4) and (-1, 3).

**Solution:** Here x$_1$ = 2, x$_2$ = -1, y$_1$ = 4 and y$_2$ = 3

Slope of line (m) =** **$\frac{y_2 - y_1}{x_2 - x_1}$

= $\frac{3-4}{-1-2}$

= $\frac{-1}{-3}$

= $\frac{1}{3}$

=> m = $\frac{1}{3}$.