In the previous section we completed the discussion on Probability. In this section we will discuss about Graphs. First we will learn the ‘method for fixing the position of points’. Let us see an example:
Rectangle ABCD in the fig.2.1 below represents the interior (portion inside the walls) of a room. The room has a length of 400 cm and a width of 300 cm. All the four corners of the room are right angled (90o).
Fig.2.1
Interior of a room
We want to fix a lamp in the room. [Assume that the lamp is a pedestal type. So that it can be placed any where on the floor. It does not need any support from walls or ceiling]. We ask the electrician to fix the lamp. The electrician will ask us where exactly do we want the lamp to be fixed?. We tell him that we want it near the corner A of the room. But look at the fig. below:
Fig.2.2
Possible positions near corner A
All the positions P1, P2 and P3 shown in the fig. are ‘near the corner A’. The position that the electrician feels is ‘near corner A’ may not be satisfactory for us. Thus it is clear that
• This information is not sufficient to fix the position of the lamp.
So we provide an additional information: The lamp should be placed ‘90 cm away from wall AB’. Now look at the fig. below:
Fig.2.3
Positions at a distance of 90 cm from wall AB
The red dashed line is parallel to wall AB, and is 90 cm away from it. So, all the positions P1, P2 and P3 shown in the fig. are ‘near the corner A’ and ‘90 cm away from wall AB’. The position that the electrician feels is ‘near corner A’ and '90 cm away from wall AB’ may not be satisfactory for us. So it is clear that
• This additional information is also not sufficient to fix the position of the lamp.
So we provide one more information: The lamp should be placed ‘40 cm away from wall AD’. Now look at the fig. below:
Fig.2.4
Position at a distance of 90 cm from AB and 40 cm from AD
The blue dashed line is parallel to wall AD, and is 40 cm away from it. The lamp placed any where on this line will satisfy our new condition. But we have given another condition previously: It should be 90 cm away from AB. There is only one position which will satisfy both these condition. It is the point of intersection of the red and blue dashed lines.
So with the two conditions: (1) 90 cm from AB and (2) 40 cm from AD, we fixed the position of the lamp. There is no need for the first condition that ‘it should be near corner A’. In fact, vague information like ‘near’ are not used in mathematics. We use only exact quantities.
One important point to note is that, the measurements should be exactly ‘parallel’ or ‘perpendicular’ which ever is applicable. This can be explained as follows:
In fig.2.4, we know that the red dashed line is exactly parallel to wall AB. And it is at a distance of 90 cm from AB. So this 90 cm should be measured in an exact perpendicular direction to wall AB. Other wise, the parallel line will be parallel, but at some other distance from AB. This is illustrated in the fig.2.5 below:
Fig.2.5
Error in position when measurement is not perpendicular
The magenta coloured arrow shows the measurement of 90 cm in an inclined direction. It is not perpendicular to AB. As a result, we will get a red dotted parallel line which is below the actual required red dashed parallel line. The dotted line is not at an exact parallel distance of 90 cm, and hence it is a wrong line. So it is important to measure the distance in an exact perpendicular direction from AB. Similarly, 40 cm should be measured in an exact perpendicular direction from wall AD.
In the next section we will discuss how the final position of the lamp can be shown on a sheet of paper.
Rectangle ABCD in the fig.2.1 below represents the interior (portion inside the walls) of a room. The room has a length of 400 cm and a width of 300 cm. All the four corners of the room are right angled (90o).
Fig.2.1
Interior of a room
Fig.2.2
Possible positions near corner A
All the positions P1, P2 and P3 shown in the fig. are ‘near the corner A’. The position that the electrician feels is ‘near corner A’ may not be satisfactory for us. Thus it is clear that
• This information is not sufficient to fix the position of the lamp.
So we provide an additional information: The lamp should be placed ‘90 cm away from wall AB’. Now look at the fig. below:
Fig.2.3
Positions at a distance of 90 cm from wall AB
• This additional information is also not sufficient to fix the position of the lamp.
So we provide one more information: The lamp should be placed ‘40 cm away from wall AD’. Now look at the fig. below:
Fig.2.4
Position at a distance of 90 cm from AB and 40 cm from AD
So with the two conditions: (1) 90 cm from AB and (2) 40 cm from AD, we fixed the position of the lamp. There is no need for the first condition that ‘it should be near corner A’. In fact, vague information like ‘near’ are not used in mathematics. We use only exact quantities.
One important point to note is that, the measurements should be exactly ‘parallel’ or ‘perpendicular’ which ever is applicable. This can be explained as follows:
In fig.2.4, we know that the red dashed line is exactly parallel to wall AB. And it is at a distance of 90 cm from AB. So this 90 cm should be measured in an exact perpendicular direction to wall AB. Other wise, the parallel line will be parallel, but at some other distance from AB. This is illustrated in the fig.2.5 below:
Fig.2.5
Error in position when measurement is not perpendicular
In the next section we will discuss how the final position of the lamp can be shown on a sheet of paper.
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