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(1) Number line : It is a line on which numbers are marked at definite intervals.
For Example:On the number line, distances from a fixed point are marked in equal unitsâ€™ positively in one direction and negatively in the other direction.
Origin : It is the point from which the distances are marked in positive and negative direction. In the figure, 0 denotes the origin.
(2) Cartesian System :Â It is the system of fixing a point with the help of two measurements which are vertical and horizontal lines.(i) The figure above shows a Cartesian system formed by combining a vertical and horizontal line.
(ii) The horizontal line Xâ€™X is called the x-axis and the vertical line YYâ€™ is called the y-axis.
(iii) Origin : It is the point where Xâ€™X and Yâ€™Y intersect each other. It is denoted by O.
(iv) The positive numbers lie on the directions OX and OY, hence, OX and OY are called the positive directions of the x-axis and y-axis respectively. And the negative numbers lie on the directions OXâ€™ and OYâ€™, hence, OXâ€™ and OYâ€™ are called the negative directions of the x-axis and y-axis respectively.(v) Here, the axes divide the plane into four parts. These four parts are called the quadrants.
(vi) Relationship between signs of coordinates of a point and the quadrant of a point:
(a) If a point is in the 1st quadrant, then the point will be in the form (+, +), since the 1st quadrant is enclosed by the positive x - axis and the positive y - axis.
(b) If a point is in the 2nd quadrant, then the point will be in the form (â€“, +), since the 2nd quadrant is enclosed by the negative x - axis and the positive y - axis.
(c) If a point is in the 3rd quadrant, then the point will be in the form (â€“, â€“), since the 3rd quadrant is enclosed by the negative x - axis and the negative y - axis.
(d) If a point is in the 4th quadrant, then the point will be in the form (+, â€“), since the 4th quadrant is enclosed by the positive x - axis and the negative y â€“ axis.
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(vii) The plane is also called as Cartesian plane or coordinate plane or xy-plane. And the axes are called the coordinate axes.
(3) Co-ordinates of a point:
(i) Consider the figure given below to understand the coordinates of a point.(ii) For the given point A, it is observed that it is obtained after moving 2 units on x-axis and 3 units on y-axis.
(iii) X â€“ coordinate: It is the perpendicular distance from y-axis measured along the x-axis. For the point A, it is +2. It is also called the abscissa.
(iv) Y â€“ coordinate: It is the perpendicular distance from x-axis measured along the y-axis. For the point A, it is +3. It is also called the ordinate.
(v) The coordinates are stated in the brackets, the x-coordinate comes first then the y-cordinate. Hence, coordinates of point A are (2, 3).
(vi) The coordinates of origin are (0, 0). It has zero distance from both the axes so that its abscissa and ordinate are both zero.
Note: The coordinates of each point in Cartesian system is unique.
For example:Â Write the coordinates of the points marked on the axes.From the figure,
(i) The coordinates of point A are (4, 0).
(ii) The coordinates of point B are (0, 3).
(iii) The coordinates of point C are (-5, 0).
(iv) The coordinates of point D are (0, -4).
(v) The coordinates of point E are (2/3, 0).
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(4) Plotting a Point in the Plane if its Coordinates are given:
To plot any given point (a, b), the distance of this point from the y - axis along the positive x - axis is â€˜aâ€™ units and the distance of the point from the x - axis along the positive y - axis is â€˜bâ€™ units. Hence, starting from the origin O, we count â€˜aâ€™ units on the positive x - axis and mark the corresponding point. And starting from this point, we move in the positive direction of the y - axis and count b units and mark the corresponding point.
For example:Â Locate points (5, 0), (0, 5), (5, 2), (-3, 5), (-3, -5) and (5, -3) in the Cartesian plane.
(i) The location of given different points is shown in the Cartesian plane below.
(ii) Here, scale is 1 cm = 1 unit for both x-axis and y-axis.
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