Distance between two points is the length of the line segment that connects the two given points. Distance between two points in coordinate geometry can be calculated by finding the length of the line segment joining the given coordinates. Let us understand the formula to find the distance between two points in a twodimensional and threedimensional plane.
The distance between any two points is the length of the line segment joining the points. There is only one line passing through two points. So, the distance between two points can be calculated by finding the length of this line segment connecting the two points. For example, if A and B are two points and if AB =10 cm, it means that the distance between A and B is 10 cm.
The distance between two points is the length of the line segment joining them (but this CANNOT be the length of the curve joining them). Note that the distance between two points is always positive.
The distance between two points using the given coordinates can be calculated by applying the distance formula. For any point given in the 2D plane, we can apply the 2D distance formula or the Euclidean distance formula given as,
Formula for Distance Between Two Points:
The formula for the distance, d d, between two points whose coordinates are (x_{1}, y_{1}) and (x_{2}, y_{2}) is:
d = √[(x_{2} − x_{1})^{2} + (y_{2 }− y_{1})^{2}]
This is called the Distance Formula.
To find the distance between two points given in 3D plane, we can apply the 3D distance formula, given as,
d = √[(x_{2 }− x_{1})^{2} + (y_{2} − y_{1})^{2 }+ (z_{2} − z_{1})^{2}]
To derive the formula to calculate the distance between two points in a twodimensional plane, let us assume that there are two points with the coordinates given as, A(x_{1}, y_{1}) B(x_{2}, y_{2})
Next, we will assume that the line segment joining A and B is AB = d. Now, we will plot the given points on the coordinate plane and join them by a line.
Next, we will construct a rightangled triangle with AB as the hypotenuse.
Applying Pythagoras theorem for the △ABC:
AB^{2} = AC^{2} + BC^{2}
d2 = ( x_{2} − x_{1} )^{2} + ( y_{2} − y_{1} )^{2} (Values from the figure)
Here, the vertical distance between the given points is y_{2} − y_{1}.
The horizontal distance between the given points is x_{2} − x_{1}.
d = √[(x_{2} − x_{1})^{2 }+ (y_{2} − y_{1})^{2}] (Taking square root on both sides)
Thus, the distance formula to find the distance between two points is proved.
Note: In case the two points A and B are on the xaxis, i.e. the coordinates of A and B are (x_{1}, 0) and (x_{2}, 0) respectively, then the distance between two points AB = x_{2} − x_{1}.
Using similar steps and concept, we can also derive the formula to find the distance between two points given in the 3D plane.
The distance between two points using the given coordinates can be calculated with the help of the following given steps:
Note: We can apply the 3D distance formula in case the two points are given in 3D plane, d = √[(x_{2 }− x_{1})^{2} + (y_{2} − y_{1})^{2} + (z_{2} − z_{1})^{2}]
The distance between two points in a complex plane or two complex numbers z_{1} = a + ib and z_{2} = c + id in the complex plane is the distance between points (a, b) and (c, d), given as,
z_{1} − z_{2} = √[(a − c)^{2 }+ (b − d)^{2}]
406 videos217 docs164 tests


Explore Courses for Class 10 exam
