# Introduction

It was the work of the French mathematician Rene Descartes (1596-1650) that merged algebra and geometry into a single unified subject. The genius of Descartes lay in his idea of a co-ordinate system. In such a system every point (a geometric concept) is assigned a pair of numbers (an arithmetic concept) as its unique "address". Descartesâ€™ idea is based on two real lines intersecting at right angles, as shown in figure below. The horizontal real line is usually called x-axis and the vertical line is usually called y-axis. The point of intersection of these two axes is called the origin and is labelled O on each number line. The axes divide the plane into four parts called quadrants. The plane itself is called the co-ordinate plane or the Cartesian plane or xy-plane.

The positive direction on the x-axis is to the right and the positive direction on the y-axis is upward. The negative direction of x-axis is to the left and negative direction of y-axis is downwards. Each point in the plane corresponds to an ordered pair (x, y) of real numbers x and y, called co-ordinates of the point.

The x-coordinate (or abscissa) represents the directed distance from y-axis to the point, and the y-coordinate (or ordinate) represents the directed distance from x-axis to the point.

In this figure note that (2, 5) lies in the first quadrant, (-4, 6) in the second quadrant, (-5, -5) in the third quadrant and (3, -3) in the fourth quadrant.

Note that if a point lies on x-axis, its y-co-ordinate is zero. For instance, see the points (âˆ’6, 0) and (7, 0) in the figure. Also, note that if a point lies on y-axis, then its x-co-ordinate is zero. For instance see the points (0, âˆ’6) and (0, 7) in the figure above.

The origin O has co-ordinates (0, 0). Now note that if two points A and B have the same y-co-ordinates, A and B must lie on the same horizontal line. For instance see points (0, âˆ’3) and (3, âˆ’3) in the figure above. Note that if the two points P and Q have the same x co-ordinates, P and Q must lie on the same vertical line. For instance see points (âˆ’4, 6) and (âˆ’4, âˆ’2) in the figure above.

We know that if d is the distance between two points a and b on the number line, d = |a â€“ b|.

The same rule is used to find the distance between two points lying on the same horizontal or vertical line in the plane. Therefore, if two points A and B lie on a horizontal line, the distance between them can be obtained by taking the absolute value of the difference of the x-coordinate of B and the x-coordinate of A. Thus if x-coordinate of A is x_{1} and x-coordinate of B is x_{2}, the distance between A and B is given by |x_{2} âˆ’ x_{1}|. In symbols, d (A, B) = |x_{2 } âˆ’ x_{1}|. Thus, the distance between (âˆ’4, 2) and (5, 2) is |5 â€“ (âˆ’4)| = 9.

Similarly, if two points P (x_{1}, y_{1}) and Q (x_{1}, y_{2}) lie on the same vertical line, the distance d (P, Q) between P and Q is given by |y_{2} â€“ y_{1}|.

We now consider the case when we drop the restriction that the line segment is either horizontal or vertical. To develop the formula, we use Pythagorean Theorem, which says that for a right-angled triangle with hypotenuse c and sides a and b, we have the relationship a^{2} + b^{2} = c^{2} as shown in the figure.

(The converse is also true. That is, if a^{2} + b^{2} = c^{2}, the triangle is a right-angled triangle.)

Using this theorem we can find the distance between any two points ( x_{1}, y_{1}) and (x_{2}, y_{2}).