Write the derivation of the distance formula in the coordinate geometry with the help of a diagram.
Answers
Step-by-step explanation:
In the previous example, we calculated the distance between two points in the plane with specific coordinates. Let us now generalize this. Suppose that we have two points with any arbitrary coordinates:
A
(
x
1
,
y
1
)
B
(
x
2
,
y
2
)
How do we calculate AB? Once again, we use the Pythagoras Theorem. We complete the right triangle ABC, where C is the point with the same y coordinate as A and the same x coordinate as B. Consider the following diagram:
Straight line distance and Pythagoras theorem
In this right triangle, note that AC is the difference between the x coordinates of A and C, and hence we write:
A
C
=
|
x
2
−
x
1
|
Note how we have used the modulus sign, because we want the positive value of the difference between A and C. In this particular case, C is to the right of A, and so the x coordinate of C will be larger than that of A. However, in a general situation, any of the two can be larger, and so it’s a good idea to use the modulus sign.
Similarly, BC will be equal to the difference between the y coordinates of B and C. Hence, we have:
B
C
=
|
y
2
−
y
1
|
Finally, we apply the Pythagoras Theorem to calculate AB:
A
B
=
√
A
C
2
+
B
C
2
=
√
|
x
2
−
x
1
|
2
+
|
y
2
−
y
1
|
2
⇒
A
B
=
√
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
This is the widely used distance formula to determine the distance between any two points in the coordinate plane. Note that in the final expression, we removed the modulus signs, since the terms got squared – so it doesn’t matter whether the original terms are negative or positive. For example, the square of 2 is 4, the same as the square of –2.
