In the previous section we learned the details about the Cartesian coordinate system. Let us now see some numerical examples.
Solved example 2.1
Plot the point A (5,7) and another point B (7,5)
Solution:
First we draw the axes to a suitable scale. Here the maximum value is 7. It can be easily accommodated in an ordinary Graph sheet. So we will use as scale of 1:1.
Plotting A: From the origin O, we move 5 units to the right in the direction of the X-axis (because 5 is the x- coordinate). From there we move 7 units upwards in the direction of the Y axis (because 7 is the y- coordinate). Thus the final position of A is reached. B can be plotted in the same way. Both are shown in the fig.2.10.
Solved example 2.2
Four points A, B, C and D are marked on a Graph sheet shown in fig.2.11. Find the coordinates of each of these points.
Solution:
Point A: In order to find the coordinates, we ask ourselves the question: Which route should we take to reach A. Let us analyse:
The beginning of the journey is from origin (0,0). From there we travel 2 units in the direction of the X axis. So the x- coordinate is 2. Then we travel 4 units in the direction of the Y axis. So the y- coordinate is 4. Thus we will reach point A, and we have the coordinates. We can write: The coordinates of A are (2,4)
Let us carefully examine the above journey that we took. We can form some general rules for the journey:
■ Every journey starts from the origin
■ The journey has two laps
■ In the first lap, we travel towards the right
■ In the second lap we travel upwards
Consider the first lap. We started from the origin and travelled in the direction of the X axis. So we have two conditions:
• Starting point → The origin
• Direction → The direction of the X axis
There is only one route that will satisfy both the above conditions simultaneously. That route is the X axis itself. Because the X axis passes through the origin. So, in the first lap, we travelled directly above the X axis.
Consider the second lap. We started from ‘the mark which indicates 2 on the X axis’ and travelled in the direction of the Y axis. So here also we have two conditions:
• Starting point → The mark which indicates 2 on the X axis
• Direction → The direction of the y axis
There is only one route that will satisfy both the above conditions simultaneously. That route is a line parallel to the Y axis. And this line is 2 units away from the Y axis. Because we have already moved away from the Y axis in the first lap. So, in the second lap, we travelled parallel to the Y axis. Not above it.
In this way, the exact route that we take, and the distance travelled in each lap should be carefully considered while fixing the coordinates of a given point OR fixing the position of a point with given coordinates.
From the above discussion we can say that:
• In the first lap, we travel away from the Y axis.
• The point that we reach at the end of the first lap, is the x- coordinate
• So x- coordinate is the distance away from the Y axis
• In the scond lap, we travel away from the X axis.
• The point that we reach at the end of the second lap, is the final point
• The distance that we have to travel in the second lap depends on the y-coordinate
• So y- coordinate is the distance away from the X axis
Point B: Based on the above discussion, we can say that the first lap is 6 units and the second lap is 3 units. Thus: The coordinates of B are (6,3)
Point C: Here the point lies on the X axis. In the first lap we travel 4 units. But now we have already reached the point. We do not have to take the second lap. Not even a small fraction of 1 unit. So the distance in the second lap is zero. Thus: The coordinates of C are (4,0)
From this we can make an important conclusion: If
• There is no travel in the second lap → There is no travel parallel to the Y axis → The point lies on the X axis → The The y- coordinate of the point is ‘0’.
Point D: Here the point lies on the X axis. We begin our journey from the origin as usual. But we need not move to the right over the X axis. Not even a small fraction of one unit. So the distance travelled in the first lap is ‘0’. In the second lap, we travel 7 units in the direction of the Y axis. As we have not moved to the right in the first lap, the route of the second lap lies directly above the Y axis. The distance from the Y axis is zero. Thus: The coordinates of D are (0,7)
From this we can make an important conclusion: If
• There is no travel in the first lap → There is no travel over the X axis → The point lies on the Y axis → The The x- coordinate of the point is ‘0’.
Solved example 2.3
Plot the following points on a Graph sheet:
(a) (2,4), (4,4), (5,4), (7,4) (b) (3,1), (3,4), (3,6.5), (3,9) (c) (4,0), (7,0), (8.7,0), (10,0) (d) (2,2), (4,3), (8,5), (12,7)
Solution:
(a) The points are plotted as shown in fig.2.12(a) below. All the points lie on a line. The line is named as AB. We can see some peculiarities in the given points:
• All the points have their y- coordinates equal. It is 4
• This will mean that all the points are at the same distance (4 units) away from the X axis.
• So the line AB that is formed by the points will be parallel to the X axis
(b) The points are plotted as shown in fig.2.12(b) below. All the points lie on a line. The line is named as PQ. We can see some peculiarities in the given points:
• All the points have their x- coordinates equal. It is 3
• This will mean that all the points are at the same distance (3 units) away from the Y axis.
• So the line that is formed by the points will be parallel to the Y axis
(c) The points are plotted as shown in fig.2.12(c) below. All the points lie on a line. We can see some peculiarities in the given points:
• All the points have their y- coordinates equal. It is 0
• This will mean that all the points are at zero distance away from the X axis. Thus all the points lie on the X axis itself
• So the line that is formed by the points will lie directly over the X axis
(d) The points are plotted as shown in fig.2.12(d) below. All the points lie on a line. The line is named as XY. We can see some peculiarities in the given points:
• All the points have both their x and y- coordinates different.
• This will mean that the line formed by the points will not be parallel to X or Y axes.
In the next section we will see some more examples.
Solved example 2.1
Plot the point A (5,7) and another point B (7,5)
 |
| Fig.2.10 |
First we draw the axes to a suitable scale. Here the maximum value is 7. It can be easily accommodated in an ordinary Graph sheet. So we will use as scale of 1:1.
Plotting A: From the origin O, we move 5 units to the right in the direction of the X-axis (because 5 is the x- coordinate). From there we move 7 units upwards in the direction of the Y axis (because 7 is the y- coordinate). Thus the final position of A is reached. B can be plotted in the same way. Both are shown in the fig.2.10.
 |
| Fig.2.11 |
Solved example 2.2
Four points A, B, C and D are marked on a Graph sheet shown in fig.2.11. Find the coordinates of each of these points.
Solution:
Point A: In order to find the coordinates, we ask ourselves the question: Which route should we take to reach A. Let us analyse:
The beginning of the journey is from origin (0,0). From there we travel 2 units in the direction of the X axis. So the x- coordinate is 2. Then we travel 4 units in the direction of the Y axis. So the y- coordinate is 4. Thus we will reach point A, and we have the coordinates. We can write: The coordinates of A are (2,4)
Let us carefully examine the above journey that we took. We can form some general rules for the journey:
■ Every journey starts from the origin
■ The journey has two laps
■ In the first lap, we travel towards the right
■ In the second lap we travel upwards
Consider the first lap. We started from the origin and travelled in the direction of the X axis. So we have two conditions:
• Starting point → The origin
• Direction → The direction of the X axis
There is only one route that will satisfy both the above conditions simultaneously. That route is the X axis itself. Because the X axis passes through the origin. So, in the first lap, we travelled directly above the X axis.
Consider the second lap. We started from ‘the mark which indicates 2 on the X axis’ and travelled in the direction of the Y axis. So here also we have two conditions:
• Starting point → The mark which indicates 2 on the X axis
• Direction → The direction of the y axis
There is only one route that will satisfy both the above conditions simultaneously. That route is a line parallel to the Y axis. And this line is 2 units away from the Y axis. Because we have already moved away from the Y axis in the first lap. So, in the second lap, we travelled parallel to the Y axis. Not above it.
In this way, the exact route that we take, and the distance travelled in each lap should be carefully considered while fixing the coordinates of a given point OR fixing the position of a point with given coordinates.
From the above discussion we can say that:
• In the first lap, we travel away from the Y axis.
• The point that we reach at the end of the first lap, is the x- coordinate
• So x- coordinate is the distance away from the Y axis
• In the scond lap, we travel away from the X axis.
• The point that we reach at the end of the second lap, is the final point
• The distance that we have to travel in the second lap depends on the y-coordinate
• So y- coordinate is the distance away from the X axis
Point B: Based on the above discussion, we can say that the first lap is 6 units and the second lap is 3 units. Thus: The coordinates of B are (6,3)
Point C: Here the point lies on the X axis. In the first lap we travel 4 units. But now we have already reached the point. We do not have to take the second lap. Not even a small fraction of 1 unit. So the distance in the second lap is zero. Thus: The coordinates of C are (4,0)
From this we can make an important conclusion: If
• There is no travel in the second lap → There is no travel parallel to the Y axis → The point lies on the X axis → The The y- coordinate of the point is ‘0’.
Point D: Here the point lies on the X axis. We begin our journey from the origin as usual. But we need not move to the right over the X axis. Not even a small fraction of one unit. So the distance travelled in the first lap is ‘0’. In the second lap, we travel 7 units in the direction of the Y axis. As we have not moved to the right in the first lap, the route of the second lap lies directly above the Y axis. The distance from the Y axis is zero. Thus: The coordinates of D are (0,7)
From this we can make an important conclusion: If
• There is no travel in the first lap → There is no travel over the X axis → The point lies on the Y axis → The The x- coordinate of the point is ‘0’.
Solved example 2.3
Plot the following points on a Graph sheet:
(a) (2,4), (4,4), (5,4), (7,4) (b) (3,1), (3,4), (3,6.5), (3,9) (c) (4,0), (7,0), (8.7,0), (10,0) (d) (2,2), (4,3), (8,5), (12,7)
(a) The points are plotted as shown in fig.2.12(a) below. All the points lie on a line. The line is named as AB. We can see some peculiarities in the given points:
• All the points have their y- coordinates equal. It is 4
• This will mean that all the points are at the same distance (4 units) away from the X axis.
• So the line AB that is formed by the points will be parallel to the X axis
 |
| Fig.2.12(a) |
• All the points have their x- coordinates equal. It is 3
• This will mean that all the points are at the same distance (3 units) away from the Y axis.
• So the line that is formed by the points will be parallel to the Y axis
 |
| Fig.2.12(b) |
• All the points have their y- coordinates equal. It is 0
• This will mean that all the points are at zero distance away from the X axis. Thus all the points lie on the X axis itself
• So the line that is formed by the points will lie directly over the X axis
 |
| Fig.2.12(c) |
• All the points have both their x and y- coordinates different.
• This will mean that the line formed by the points will not be parallel to X or Y axes.
 |
| Fig.2.12(d) |
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