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it is distance formula
syam75:
it is distance formula
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In the above diagram, let’s consider 2 points A(x1,y1) and B(x2,y2). Connect the points A and B directly forming a slanting line AB. Now complete the triangle by joining the points A and B to a common point C such that AC is parallel to the y-axis while BC is parallel to the x-axis.
Now for the coordinates of point C.
Since we considered the point C to be parallel to x-axis the y-coordinate of C will be the same as the y-coordinate of B which is y2 and since C is also parallel to the y-axis the x-coordinate will be equal to the x-coordinate of A which is x1.
Therefore the coordinates of point C are C (x1,y2).
It is a well-known fact that on the x-y coordinate plane the x-axis cuts the y-axis at 90 degrees. That means both the axes are perpendicular to each other. Since the sides, AC and BC are parallel to the y-axis and the x-axis respectively, what we now have is a right-angled triangle ACB where side AB is the hypotenuse, side AC is the perpendicular and CB is the base.

In order to find the length of side AC, we need to find the distance between points A and C. As we have already established the fact that AC is parallel to the y-axis, the x-coordinates will be the same and we cannot use it to calculate the distance.
The only information we are left with are the y-coordinates. If we observe carefully, subtracting the y-coordinate values of C from the y-coordinate value of A we get the distance between the points A and C.
And that my friends, is how we found the length of side AC.
side AC= y2-y1 (1)
Similarly, we also find the length of side CB. The only difference here will be that the y-coordinates remain the same and we subtract the x-coordinates. Therefore,
side CB= x2-x1 (2)
Triangle ACB is a right-angled triangle. We need to find hypotenuse AB.
To do that we use the Pythagoras Theorem. So,
AB²=BC²+AC²
From (1) and (2),
AB²=(x2-x1)²+(y2-y1)²
Taking Square roots on both sides,
AB=√{(x2-x1)²+(y2-y1)²}
What we derived just now is the Distance formula.
So the Distance Formula is,
AB=√[(x2-x1)²+(y2-y1)²]
Now for the coordinates of point C.
Since we considered the point C to be parallel to x-axis the y-coordinate of C will be the same as the y-coordinate of B which is y2 and since C is also parallel to the y-axis the x-coordinate will be equal to the x-coordinate of A which is x1.
Therefore the coordinates of point C are C (x1,y2).
It is a well-known fact that on the x-y coordinate plane the x-axis cuts the y-axis at 90 degrees. That means both the axes are perpendicular to each other. Since the sides, AC and BC are parallel to the y-axis and the x-axis respectively, what we now have is a right-angled triangle ACB where side AB is the hypotenuse, side AC is the perpendicular and CB is the base.

In order to find the length of side AC, we need to find the distance between points A and C. As we have already established the fact that AC is parallel to the y-axis, the x-coordinates will be the same and we cannot use it to calculate the distance.
The only information we are left with are the y-coordinates. If we observe carefully, subtracting the y-coordinate values of C from the y-coordinate value of A we get the distance between the points A and C.
And that my friends, is how we found the length of side AC.
side AC= y2-y1 (1)
Similarly, we also find the length of side CB. The only difference here will be that the y-coordinates remain the same and we subtract the x-coordinates. Therefore,
side CB= x2-x1 (2)
Triangle ACB is a right-angled triangle. We need to find hypotenuse AB.
To do that we use the Pythagoras Theorem. So,
AB²=BC²+AC²
From (1) and (2),
AB²=(x2-x1)²+(y2-y1)²
Taking Square roots on both sides,
AB=√{(x2-x1)²+(y2-y1)²}
What we derived just now is the Distance formula.
So the Distance Formula is,
AB=√[(x2-x1)²+(y2-y1)²]
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