`♦` Plotting a Point in the Plane if its Coordinates are Given

Uptil now we have drawn the points for you, and asked you to give their coordinates. Now we will show you how we place these points in the plane if we know its coordinates. We call this process “plotting the point”.

Let the coordinates of a point be (3, 5). We want to plot this point in the coordinate plane.

We draw the coordinate axes, and choose our units such that one centimetre represents one unit on both the axes.

The coordinates of the point (3, 5) tell us that the distance of this point from the y - axis along the positive x - axis is 3 units and the distance of the point from the x - axis along the positive y - axis is 5 units.

Starting from the origin O, we count 3 units on the positive x - axis and mark the corresponding point as A. Now, starting from A, we move in the positive direction of the y - axis and count 5 units and mark the corresponding point as P (see Fig.3.15).

You see that the distance of P from the y - axis is 3 units and from the x - axis is 5 units. Hence, P is the position of the point. Note that P lies in the 1st quadrant, since both the coordinates of P are positive.

Similarly, you can plot the point Q (5, – 4) in the coordinate plane. The distance of Q from the x - axis is 4 units along the negative y - axis, so that its y - coordinate is – 4 (see Fig.3.15). The point Q lies in the 4th quadrant.

Let the coordinates of a point be (3, 5). We want to plot this point in the coordinate plane.

We draw the coordinate axes, and choose our units such that one centimetre represents one unit on both the axes.

The coordinates of the point (3, 5) tell us that the distance of this point from the y - axis along the positive x - axis is 3 units and the distance of the point from the x - axis along the positive y - axis is 5 units.

Starting from the origin O, we count 3 units on the positive x - axis and mark the corresponding point as A. Now, starting from A, we move in the positive direction of the y - axis and count 5 units and mark the corresponding point as P (see Fig.3.15).

You see that the distance of P from the y - axis is 3 units and from the x - axis is 5 units. Hence, P is the position of the point. Note that P lies in the 1st quadrant, since both the coordinates of P are positive.

Similarly, you can plot the point Q (5, – 4) in the coordinate plane. The distance of Q from the x - axis is 4 units along the negative y - axis, so that its y - coordinate is – 4 (see Fig.3.15). The point Q lies in the 4th quadrant.

Q 3220267111

Locate the points `(5, 0), (0, 5), (2, 5), (5, 2), (–3, 5), (–3, –5), (5, –3)` and `(6, 1)` in the Cartesian plane.

Class 9 Chapter 3 Example 3

Class 9 Chapter 3 Example 3

Taking `1` cm = `1` unit, we draw the x - axis and the y - axis. The positions of

the points are shown by dots in Fig.3.16.

Note : In the example above, you see that `(5, 0)` and `(0, 5)` are not at the same

position. Similarly, `(5, 2)` and `(2, 5)` are at different positions. Also, `(–3, 5)` and `(5, –3)`

are at different positions. By taking several such examples, you will find that, if x ≠ y,

then the position of (x, y) in the Cartesian plane is different from the position

of (y, x). So, if we interchange the coordinates x and y, the position of (y, x) will differ

from the position of (x, y). This means that the order of x and y is important in (x, y).

Therefore, (x, y) is called an ordered pair. The ordered pair (x, y) ≠ ordered pair (y, x),

if x ≠ y. Also (x, y) = (y, x), if x = y.

Q 3230267112

Plot the following ordered pairs `(x, y)` of numbers as points in the Cartesian plane. Use the scale `1` cm `= 1` unit on the axes.

Class 9 Chapter 3 Example 4

Class 9 Chapter 3 Example 4

The pairs of numbers given in the table can be represented by the points

`(– 3, 7), (0, –3.5), (– 1, – 3), (4, 4)` and `(2, – 3)`. The locations of the points are shown

by dots in Fig.3.17.

Activity 2 : A game for two persons (Requirements: two counters or coins, graph paper, two dice of different colours, say red and green):

Place each counter at (0, 0). Each player throws two dice simultaneously. When the first player does so, suppose the red die shows 3 and the green one shows 1. So, she moves her counter to (3, 1).

Similarly, if the second player throws 2 on the red and 4 on the green, she moves her counter to (2, 4). On the second throw, if the first player throws 1 on the red and 4 on the green, she moves her counter from (3, 1) to (3 + 1, 1 + 4), that is, adding 1 to the x - coordinate and 4 to the y - coordinate of (3, 1).

The purpose of the game is to arrive first at (10, 10) without overshooting, i.e., neither the abscissa nor the ordinate can be greater than 10.

Also, a counter should not coincide with the position held by another counter. For example, if the first player’s counter moves on to a point already occupied by the counter of the second player, then the second player’s counter goes to (0, 0).

If a move is not possible without overshooting, the player misses that turn. You can extend this game to play with more friends.

`"Remark :"` Plotting of points in the Cartesian plane can be compared to some extent with drawing of graphs in different situations such as Time-

Distance Graph, Side- Perimeter Graph, etc which you have come across in earlier classes. In such situations, we may call the axes, t-axis, d-axis, s-axis or p-axis, etc. in place of the x and y axes.

Place each counter at (0, 0). Each player throws two dice simultaneously. When the first player does so, suppose the red die shows 3 and the green one shows 1. So, she moves her counter to (3, 1).

Similarly, if the second player throws 2 on the red and 4 on the green, she moves her counter to (2, 4). On the second throw, if the first player throws 1 on the red and 4 on the green, she moves her counter from (3, 1) to (3 + 1, 1 + 4), that is, adding 1 to the x - coordinate and 4 to the y - coordinate of (3, 1).

The purpose of the game is to arrive first at (10, 10) without overshooting, i.e., neither the abscissa nor the ordinate can be greater than 10.

Also, a counter should not coincide with the position held by another counter. For example, if the first player’s counter moves on to a point already occupied by the counter of the second player, then the second player’s counter goes to (0, 0).

If a move is not possible without overshooting, the player misses that turn. You can extend this game to play with more friends.

`"Remark :"` Plotting of points in the Cartesian plane can be compared to some extent with drawing of graphs in different situations such as Time-

Distance Graph, Side- Perimeter Graph, etc which you have come across in earlier classes. In such situations, we may call the axes, t-axis, d-axis, s-axis or p-axis, etc. in place of the x and y axes.