ICSE Class 8 Maths Chapter 19 Coordinate System and Graphs

Coordinate System and Graphs

In previous class, you have learnt the basic terminology of coordinate geometry, plotting of points and graphs of linear equations in two variables x and y. In this chapter, we shall strengthen these concepts and introduce how to solve graphically a pair of simultaneous linear equations in two variables.

Coordinate System

Draw two number lines X'OX and Y'OY perpendicular to each other (horizontal and vertical) on a graph paper to intersect each other at the point O. Then

(i) the horizontal line X'OX is called x-axis.

(ii) the vertical line Y'OY is called y-axis.

(iii) X'OX and Y'OY taken together are called coordinate axes.

(iv) the point O is called the origin.

The configuration so formed is called a coordinate system or coordinate plane.

Coordinates of a Point

Let P be any point in the coordinate plane. From P, draw PM perpendicular to X'OX. Then

(i) OM is called x-coordinate or abscissa of P and is usually denoted by x.

(ii) MP is called y-coordinate or ordinate of P and is usually denoted by y.

(iii) x and y taken together are called coordinates of P. It is written as (x, y) or P(x, y).

Thus, corresponding to a point P in the coordinate plane, we get an ordered pair (x, y) of real numbers; conversely, corresponding to every ordered pair (x, y) of real numbers, we get a point P in the coordinate plane whose abscissa = x and ordinate = y.

For example, if the abscissa of a point P in the coordinate plane is 2 and ordinate is 3, then coordinates of P are (2, 3). So we get the ordered pair (2, 3); conversely, corresponding to the ordered pair (2, 3) we get a point in the coordinate plane whose abscissa is 2 and ordinate is 3.

Teacher's Note

Understanding coordinates helps us locate places on maps, just like GPS uses latitude and longitude to find our position.

Convention for Signs of Coordinates

(i) The x-coordinate (abscissa) of a point is positive if it is measured to the right of origin and negative if it is measured to the left of origin.

(ii) The y-coordinate (ordinate) of a point is positive if it is measured above the origin and negative if it is measured below the origin.

The coordinates of the origin are (0, 0).

For any point on x-axis, its ordinate is zero so the coordinates of any point on x-axis are (x, 0). Thus, each of the points (3, 0), (-7, 0), (0, 0) lies on x-axis.

For any point on y-axis, its abscissa is zero so the coordinates of any point on y-axis are (0, y). Thus, each of the points (0, 3), (0, -7), (0, 0) lies on y-axis.

Teacher's Note

The origin (0, 0) is like the center point on a map where all directions are measured from.

Quadrants

The two axes divide the plane into four parts called quadrants.

(i) XOY is called first quadrant. Here both x and y are positive.

(ii) X'OY is called second quadrant. Here x is negative and y is positive.

(iii) X'OY' is called third quadrant. Here both x and y are negative.

(iv) Y'OX is called fourth quadrant. Here x is positive and y is negative.

Plot the points A (4, 1), B (-2, 1), C (-3, -2) and D (3, -2). Name the figure ABCD. Find its area.

Points A, B, C, D are marked in the adjoining figure. It is easy to see that it is a parallelogram.

Area = base × height

= 6 units × 3 units

= 18 square units.

Teacher's Note

Quadrants help us understand which direction from the origin a point is located, similar to compass directions in navigation.

Exercise 19.1

State whether true or false :

(i) The point (4-5, 0) lies on x-axis.

(ii) The point (0, -4) lies on x-axis.

(iii) The point (0, -3-5) lies on y-axis.

(iv) If the point (x, y) lies on x-axis, then its abscissa is zero.

(v) If the point (x, y) lies on y-axis, then its ordinate is zero.

(vi) The point (x, y) lies on y-axis if x = 0.

(vii) The point (-3, -2) lies in the fourth quadrant.

(viii) The point (5, -3) lies in the second quadrant.

Plot the following points on the same graph paper :

(i) (-3, 5) (ii) (4, 2-5) (iii) (-1, -4) (iv) (0, -4)

(v) (3, -5) (vi) (-4-5, 0) (vii) (-4, -1) (viii) (-2, 6)

Plot the points A (1, 2), B (-4, 2), C (-4, -1) and D (1, -1). What kind of quadrilateral is ABCD? Find its area.

Plot the points A (2, 0), B (0, 5) and C (-2, 0). What kind of triangle is ABC? Find its area.

Plot a rectangle which lies in first quadrant, has origin as one vertex, is 6 units long along x-axis and 4 units long along y-axis. Give the coordinates of its vertices.

Graphs of Linear Equations

To draw the graph of a linear equation in two variables x and y, proceed as under :

(i) Make a table of values. Choose three values of x and find the corresponding values of y from the given linear equation. As far as possible, take the integral values of x.

(ii) Plot these points on the graph paper (coordinate plane).

(iii) Join any two points by a straight line and check that the third point lies on it.

Graph of a linear equation in two variables is always a straight line

Draw the graph of the equation 2x + y - 1 = 0.

The given equation is 2x + y - 1 = 0

It can be written as y = -2x + 1

When x = 0, y = -2 × 0 + 1 = 1;

when x = 1, y = -2 × 1 + 1 = -1;

when x = -1, y = -2 × (-1) + 1 = 3

Plot the points A(0, 1), B(1, -1) and C(-1, 3) on the graph paper. Join any two points by a straight line. The graph of the given equation is shown in the adjoining figure. Observe that the third point lies on the straight line.

Teacher's Note

Linear equations are used in real life when calculating bills, like the total cost of groceries based on unit prices.

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