## Midpoint Formula

In this page midpoint formula we are going to see how to find the midpoint of the line segment.

Mid point:

Midpoint is the point which is exactly in the middle of the line segment joining two points (x1,y1) and (x2,y2)

Midpoint of the line segment = (x1 + x2)/2 , (y1 + y2)/2

Here are the example problems for the concept midpoint and the application of midpoint formula

Example 1:

Find the co-ordinates of the mid point of the line segment joining the points A(-3,2) and B(7,8)

Solution:

Midpoint of the line segment = (x1 + x2)/2 , (y1 + y2)/2

Here x1=-3 ,x2 = 7, y1 = 2 and y2 = 8

applying these values in the midpoint formula we get

=   (-3 + 7) / 2 , (2 + 8)/2

=   4/2 , 10/2

=  (2,5)

The midpoint of the line segment is (2,5).

Example 2:

The center of a circle is (-6,4). A diameter of the circle has its one end at the origin.Find the other end.

Solution: Let (a,b) be the required other end of the diameter.The center point is exactly middle in the diameter.

Here we have x1= 0 , x2 = a , y1 = 0 and y2 = b

Midpoint is (-6,4)

Midpoint = (x1+x2)/2 ,(y1+y2)/2

(-6,4) = (0 + a)/2 , (0 + b)/2

(-6,4) = a/2 , b/2

Equating the x and y coordinates

a/2 = -6             b/2 = 4

a = -6 x 2          b = 4 x 2

a = -12             b = 8

So the required point is (-12,8)

Example 2:

If the points A(2,-2) B(8,4) C(5,7) are the three vertices of a parallelogram ABCD taken in order,find the fourth vertex D.

Solution:

Let (a,b) be the required vertex D.

In any parallelogram the diagonals AC and BD bisect each other.That is the midpoint of the diagonal AC will be equal to the midpoint of the diagonal BD

Midpoint of the diagonal AC = (x1+x2)/2 , (y1+y2)/2

A(2,-2) C(5,7)

x1 = 2, x2 = 5 ,y1 = -2 and y2 = 7

= (2 + 5)/2 , (-2+7)/2

= 7/2 ,5/2

Midpoint of the diagonal BD =  (x1+x2)/2 , (y1+y2)/2

B(8,4) D(a,b)

x1 = 8,x2 = a, y1 = 4 and y2= b

= (8 + a)/2 , (-4+b)/2

midpoint of the diagonal AC = midpoint of the diagonal BD

(7/2,5/2) =  (8+a)/2,(-4+b)/2

(8+a)/2 =  7/2                (-4+b)/2 = 5/2

8+a = 7                      -4 + b = 5

a = 7-8                         b = 5 + 4

a = -1                           b = 9

So the required vertex is (-1,9)

Related Topics

midpoint formula to Analytical Geometry 