In the previous section we saw a few solved examples which demonstrated some simple practical applications of the Cartesian plane. In this section we will see the method of calculating the distance between any two given points in the Cartesian plane.
Consider two points P and Q on the x-axis in the fig.31.33 below:
1. The distance from origin to P is 2 units
• The distance from origin to Q is 5 units
• So distance between P and Q is (5-2) = 3 units
2. Can we find this distance using coordinates? Let us try:
(i) The coordinates of P are (2,0) and those of Q are (5,0)
(ii) The x coordinate of P is 2 and that of Q is 5
(iii) If we consider P as the first point and Q as the second point, we can write: x1 = 2 and x2 = 5
(iv) Now we can write: (x2 - x1) = (5 - 2) = 3. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
Another example:
In fig.31.34 below, both P and Q are on the negative side OX' of the x-axis
We want the distance between P and Q
1. The distance from origin to P is 5 units
• The distance from origin to Q is 2 units
• So the distance between P and Q is (5-2) = 3 units
2. Can we find this distance using coordinates? Let us try:
(i) The coordinates of P are (-5,0) and those of Q are (-2,0)
(ii) The x coordinate of P is -5 and that of Q is -2
(iii) If we consider P as the first point and Q as the second point, we can write: x1 = -5 and x2 = -2
(iv) Now we can write: (x2 - x1) = [-2 -(-5)] = [-2+5] = 3. This is the same result as above.
• So we subtracted the 'first x coordinate' from the second x coordinate'.
3. In this example, we took P as the first point and Q as the second point. What if this order is reversed?
• Let us take Q as the first point and P as the second point
(i) Then we can write: x1 = -2 and x2 = -5
(ii) Now we get: (x2 - x1) = [-5 -(-2)] = [-5+2] = -3
(iii) But 'distance' do not need a sign. This is because, it has only magnitude but no direction.
• So we can take the absolute value. We get:
• Distance between P and Q is |-3| = 3. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
■ So we find that, order of taking the given points is not important. In other words:
• When two points are given, we can take any one of them as the first point and the other as second
Another example:
In fig.31.35 below, P is on the negative side OX' and Q is on the positive side OX
We want the distance between P and Q
1. The distance from origin to P is 2 units
• The distance from origin to Q is 5 units
• Since the points are on the opposite sides, we have to add:
• The distance between P and Q is (2+5) = 7 units
2. Can we find this distance using coordinates? Let us try:
(i) The coordinates of P are (-2,0) and those of Q are (5,0)
(ii) The x coordinate of P is -2 and that of Q is 5
(iii) If we consider P as the first point and Q as the second point, we can write x1 = -2 and x2 = 5
(iv) Now we can write: (x2 - x1) = [5 -(-2)] = [5+2] = 7. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
3. In this example, we took P as the first point and Q as the second point. What if this order is reversed?
• Let us take Q as the first point and P as the second point
(i) Then we can write: x1 = 5 and x2 = -2
(ii) Now we get: (x2 - x1) = [-2 -5] = -7
(iii) But 'distance' do not need a sign. This is because, it has only magnitude but no direction.
• So we can take the absolute value. We get:
• Distance between P and Q is |-7| = 7. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
■ So here also we find that, order of taking the given points is not important. In other words:
• When two points are given, we can take any one of them as the first point and the other as second
• In the above discussion, we considered a particular case. We will write it as: 'Case 1'
• This case 1 has three sub categories. We will write it as (a), (b) and (c). So we get:
■ Case 1: The two given points lie on the x-axis
(a) Both the given points are on the positive side of the x-axis
(b) Both the given points are on the negative side of the x-axis
(c) One point is on the positive side and the other is on the negative side
• (a), (b) and (c) are the only three possibilities under Case 1
• We saw that the method of calculation is same for all. So we do not need to write each of them separately. We can write:
Calculation of distance between two given points
Case 1: The two given points lie on the x axis
• When the points lie on the x-axis, their general form can be written as: (x1,0) and (x2,0)
♦ Distance between them = |(x2-x1)|
Now we will consider the next case
Case 2: The two given points lie on a line parallel to the x-axis
Here also, there are three and only three possibilities:
(a) Both the given points are on the positive side of the x-axis
(b) Both the given points are on the negative side of the x-axis
(c) One point is on the positive side and the other is on the negative side
• Examples for all the three possibilities are shown in a single fig.31.36 below
Let us consider each possibility:
(a) Both points on the positive side of the x-axis:
(i) Consider P and Q
• The distance between them is obviously 4 units
(ii) Can we obtain this distance using coordinates? Let us try:
• Taking P as the first point and Q as the second, we get: x1 = 3 and x2 = 7
Distance = |(x2-x1)| = |(7-3)| = 4
• Taking Q as the first point and P as the second, we get: x1 = 7 and x2 = 3
Distance = |(x2-x1)| = |(3-7)| = |-4| = 4
• All results are the same
(b) Both points on the positive side of the x-axis:
(i) Consider M and N
• The distance between them is obviously 3 units
(ii) Can we obtain this distance using coordinates? Let us try:
• Taking M as the first point and N as the second, we get: x1 = -5 and x2 = -2
Distance = |(x2-x1)| = |[(-2-(-5)]| = |[-2+5]| = |[3]| = 3
• Taking N as the first point and M as the second, we get: x1 = -2 and x2 = -5
Distance = |(x2-x1)| = |[(-5-(-2)]| = |[-5+2]| = |[-3]| = 3
• All results are the same
(c) One point is on the positive side and the other is on the negative side
(i) Consider M and P
• The distance between them is obviously 8 units
(ii) Can we obtain this distance using coordinates? Let us try:
• Taking M as the first point and P as the second, we get: x1 = -5 and x2 = 3
Distance = |(x2-x1)| = |[(3-(-5)]| = |[3+5]| = |[8]| = 8
• Taking P as the first point and M as the second, we get: x1 = 3 and x2 = -5
Distance = |(x2-x1)| = |[(-5-(3)]| = |[-5-3]| = |[-8]| = 8
• All results are the same
We get '|(x2-x1)|' for (a), (b) and (c) of Case 2. We can add this to information to case 1:
Calculation of distance between two given points
Case 1: The two given points lie on the x-axis
• When the points lie on the x-axis, their general form can be written as: (x1,0) and (x2,0)
♦ Distance between them = |(x2-x1)|
Case 2: The two given points lie on a line parallel to the x-axis
• When the points lie on a line parallel to the x-axis, their general form can be written as: (x1,y) and (x2,y)
['y' is same for both points because, all points on a line parallel to the x-axis will have the same y coordinate]
♦ Distance between them = |(x2-x1)|
Now we will consider the next case
Case 3: The two given points lie on the y-axis
This is a case similar to Case 1. The result is obvious. We will get: '|(y2-y1)|'
However, the reader may take different points on the y-axis and write all the steps in his/her own notebooks
We can write the next case also easily:
Case 4: The two given points lie on a line parallel to the y-axis
This is a case similar to Case 2. The result is obvious. We will get: '|(y2-y1)|'
However, the reader may take different points on a line parallel to the y-axis and write all the steps in his/her own notebooks
We will write the four cases together:
Calculation of distance between two given points
Case 1: The two given points lie on the x-axis
• When the points lie on the x-axis, their general form can be written as: (x1,0) and (x2,0)
♦ Distance between them = |(x2-x1)|
Case 2: The two given points lie on a line parallel to the x-axis
• When the points lie on a line parallel to the x-axis, their general form can be written as: (x1,y) and (x2,y)
['y' is same for both points because, all points on a line parallel to the x-axis will have the same y coordinate]
♦ Distance between them = |(x2-x1)|
Case 3: The two given points lie on the y-axis
• When the points lie on the y-axis, their general form can be written as: (0,y1) and (0,y2)
♦ Distance between them = |(y2-y1)|
Case 4: The two given points lie on a line parallel to the y-axis
• When the points lie on a line parallel to the y-axis, their general form can be written as: (0,y1) and (0,y2)
['x' is same for both points because, all points on a line parallel to the y-axis will have the same x coordinate]
♦ Distance between them = |(y2-y1)|
Consider two points P and Q on the x-axis in the fig.31.33 below:
 |
| Fig.32.33 |
• The distance from origin to Q is 5 units
• So distance between P and Q is (5-2) = 3 units
2. Can we find this distance using coordinates? Let us try:
(i) The coordinates of P are (2,0) and those of Q are (5,0)
(ii) The x coordinate of P is 2 and that of Q is 5
(iii) If we consider P as the first point and Q as the second point, we can write: x1 = 2 and x2 = 5
(iv) Now we can write: (x2 - x1) = (5 - 2) = 3. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
Another example:
In fig.31.34 below, both P and Q are on the negative side OX' of the x-axis
 |
| Fig.31.34 |
1. The distance from origin to P is 5 units
• The distance from origin to Q is 2 units
• So the distance between P and Q is (5-2) = 3 units
2. Can we find this distance using coordinates? Let us try:
(i) The coordinates of P are (-5,0) and those of Q are (-2,0)
(ii) The x coordinate of P is -5 and that of Q is -2
(iii) If we consider P as the first point and Q as the second point, we can write: x1 = -5 and x2 = -2
(iv) Now we can write: (x2 - x1) = [-2 -(-5)] = [-2+5] = 3. This is the same result as above.
• So we subtracted the 'first x coordinate' from the second x coordinate'.
3. In this example, we took P as the first point and Q as the second point. What if this order is reversed?
• Let us take Q as the first point and P as the second point
(i) Then we can write: x1 = -2 and x2 = -5
(ii) Now we get: (x2 - x1) = [-5 -(-2)] = [-5+2] = -3
(iii) But 'distance' do not need a sign. This is because, it has only magnitude but no direction.
• So we can take the absolute value. We get:
• Distance between P and Q is |-3| = 3. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
■ So we find that, order of taking the given points is not important. In other words:
• When two points are given, we can take any one of them as the first point and the other as second
Another example:
In fig.31.35 below, P is on the negative side OX' and Q is on the positive side OX
 |
| Fig.31.35 |
1. The distance from origin to P is 2 units
• The distance from origin to Q is 5 units
• Since the points are on the opposite sides, we have to add:
• The distance between P and Q is (2+5) = 7 units
2. Can we find this distance using coordinates? Let us try:
(i) The coordinates of P are (-2,0) and those of Q are (5,0)
(ii) The x coordinate of P is -2 and that of Q is 5
(iii) If we consider P as the first point and Q as the second point, we can write x1 = -2 and x2 = 5
(iv) Now we can write: (x2 - x1) = [5 -(-2)] = [5+2] = 7. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
3. In this example, we took P as the first point and Q as the second point. What if this order is reversed?
• Let us take Q as the first point and P as the second point
(i) Then we can write: x1 = 5 and x2 = -2
(ii) Now we get: (x2 - x1) = [-2 -5] = -7
(iii) But 'distance' do not need a sign. This is because, it has only magnitude but no direction.
• So we can take the absolute value. We get:
• Distance between P and Q is |-7| = 7. This is the same result as above
• So we subtracted the 'first x coordinate' from the second x coordinate'.
■ So here also we find that, order of taking the given points is not important. In other words:
• When two points are given, we can take any one of them as the first point and the other as second
• In the above discussion, we considered a particular case. We will write it as: 'Case 1'
• This case 1 has three sub categories. We will write it as (a), (b) and (c). So we get:
■ Case 1: The two given points lie on the x-axis
(a) Both the given points are on the positive side of the x-axis
(b) Both the given points are on the negative side of the x-axis
(c) One point is on the positive side and the other is on the negative side
• (a), (b) and (c) are the only three possibilities under Case 1
• We saw that the method of calculation is same for all. So we do not need to write each of them separately. We can write:
Calculation of distance between two given points
Case 1: The two given points lie on the x axis
• When the points lie on the x-axis, their general form can be written as: (x1,0) and (x2,0)
♦ Distance between them = |(x2-x1)|
Now we will consider the next case
Case 2: The two given points lie on a line parallel to the x-axis
Here also, there are three and only three possibilities:
(a) Both the given points are on the positive side of the x-axis
(b) Both the given points are on the negative side of the x-axis
(c) One point is on the positive side and the other is on the negative side
• Examples for all the three possibilities are shown in a single fig.31.36 below
 |
| Fig.31.36 |
(a) Both points on the positive side of the x-axis:
(i) Consider P and Q
• The distance between them is obviously 4 units
(ii) Can we obtain this distance using coordinates? Let us try:
• Taking P as the first point and Q as the second, we get: x1 = 3 and x2 = 7
Distance = |(x2-x1)| = |(7-3)| = 4
• Taking Q as the first point and P as the second, we get: x1 = 7 and x2 = 3
Distance = |(x2-x1)| = |(3-7)| = |-4| = 4
• All results are the same
(b) Both points on the positive side of the x-axis:
(i) Consider M and N
• The distance between them is obviously 3 units
(ii) Can we obtain this distance using coordinates? Let us try:
• Taking M as the first point and N as the second, we get: x1 = -5 and x2 = -2
Distance = |(x2-x1)| = |[(-2-(-5)]| = |[-2+5]| = |[3]| = 3
• Taking N as the first point and M as the second, we get: x1 = -2 and x2 = -5
Distance = |(x2-x1)| = |[(-5-(-2)]| = |[-5+2]| = |[-3]| = 3
• All results are the same
(c) One point is on the positive side and the other is on the negative side
(i) Consider M and P
• The distance between them is obviously 8 units
(ii) Can we obtain this distance using coordinates? Let us try:
• Taking M as the first point and P as the second, we get: x1 = -5 and x2 = 3
Distance = |(x2-x1)| = |[(3-(-5)]| = |[3+5]| = |[8]| = 8
• Taking P as the first point and M as the second, we get: x1 = 3 and x2 = -5
Distance = |(x2-x1)| = |[(-5-(3)]| = |[-5-3]| = |[-8]| = 8
• All results are the same
We get '|(x2-x1)|' for (a), (b) and (c) of Case 2. We can add this to information to case 1:
Calculation of distance between two given points
Case 1: The two given points lie on the x-axis
• When the points lie on the x-axis, their general form can be written as: (x1,0) and (x2,0)
♦ Distance between them = |(x2-x1)|
Case 2: The two given points lie on a line parallel to the x-axis
• When the points lie on a line parallel to the x-axis, their general form can be written as: (x1,y) and (x2,y)
['y' is same for both points because, all points on a line parallel to the x-axis will have the same y coordinate]
♦ Distance between them = |(x2-x1)|
Now we will consider the next case
Case 3: The two given points lie on the y-axis
This is a case similar to Case 1. The result is obvious. We will get: '|(y2-y1)|'
However, the reader may take different points on the y-axis and write all the steps in his/her own notebooks
We can write the next case also easily:
Case 4: The two given points lie on a line parallel to the y-axis
This is a case similar to Case 2. The result is obvious. We will get: '|(y2-y1)|'
However, the reader may take different points on a line parallel to the y-axis and write all the steps in his/her own notebooks
We will write the four cases together:
Calculation of distance between two given points
Case 1: The two given points lie on the x-axis
• When the points lie on the x-axis, their general form can be written as: (x1,0) and (x2,0)
♦ Distance between them = |(x2-x1)|
Case 2: The two given points lie on a line parallel to the x-axis
• When the points lie on a line parallel to the x-axis, their general form can be written as: (x1,y) and (x2,y)
['y' is same for both points because, all points on a line parallel to the x-axis will have the same y coordinate]
♦ Distance between them = |(x2-x1)|
Case 3: The two given points lie on the y-axis
• When the points lie on the y-axis, their general form can be written as: (0,y1) and (0,y2)
♦ Distance between them = |(y2-y1)|
Case 4: The two given points lie on a line parallel to the y-axis
• When the points lie on a line parallel to the y-axis, their general form can be written as: (0,y1) and (0,y2)
['x' is same for both points because, all points on a line parallel to the y-axis will have the same x coordinate]
♦ Distance between them = |(y2-y1)|
In the next section we will see how to calculate the distance between any two points which are not on the axes or on parallel lines.
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