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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". Descartes’ 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 (−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, −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, −3) and (3, −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 (−4, 6) and (−4, −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 – 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 x1 and x-coordinate of B is x2, the distance between A and B is given by |x2 − x1|. In symbols, d (A, B) = |x2 − x1|. Thus, the distance between (−4, 2) and (5, 2) is |5 – (−4)| = 9.

Similarly, if two points P (x1, y1) and Q (x1, y2) lie on the same vertical line, the distance d (P, Q) between P and Q is given by |y2 – y1|.

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 a2 + b2 = c2 as shown in the figure.


(The converse is also true. That is, if a2 + b2 = c2, the triangle is a right-angled triangle.)

Using this theorem we can find the distance between any two points ( x1, y1) and (x2, y2).

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