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Midpoint of an Interval

Midpoint of an interval, i.e. two points A (x1, y1) and B (x2, y2) is obtained by finding the mid value midpoint_of_an_intervalalong x and y coordinates separately.

So the midpoint along the x axis = {x_1~+~x_2}/2 and the midpoint along the y axis = {y_1~+~y_2}/2.

Therefore midpoint of the line AB is ({x_1~+~x_2}/2~,~{y_1~+~y_2}/2)

 

Examples of midpoint of an interval

 

Example 1: Find the midpoint of A (-3, 2) and B (3, 5)

Midpoint of AB  =  ({x_1~+~x_2}/2~,~{y_1~+~y_2}/2)

=  ({-3~+~3}/2~,~{2~+~5}/2)

=  (0~,~7/2)

 

Example 2: ABCD is a quadrilateral with coordinates A(1, -1), B(3, 5), C(7, 7) and D(5, 1). Find the midpoints of AC and BD to prove ABCD is a parallelogram.

Midpoint of AC  =  ({x_1~+~x_2}/2~,~{y_1~+~y_2}/2)

=  ({1~+~7}/2~,~{-1~+~7}/2)

=  (4~,~3)

Midpoint of BD  =  ({x_1~+~x_2}/2~,~{y_1~+~y_2}/2)

=  ({3~+~5}/2~,~{5~+~1}/2)

=  (4~,~3)

The midpoints off the diagonals AC and BD are both (4, 3). Hence the diagonals bisect each other. Therefore ABCD is a parallelogram.

 

Example 3: The point M(3, 5) is the midpoint of the interval AB where A is the point (-1, 2). Find the coordinates of B.

coordinates_for_midpoint

Here we’ve been given the midpoint and one of the coordinates, and need to find the other coordinate.

Midpoint of an interval = ({x_1~+~x_2}/2~,~{y_1~+~y_2}/2)

3 = {-1~+~x_2}/2

6 = -1 + x2

x2 = 7

5 = {2~+~y_2}/2

10 = 2 + y2

y2 = 10 – 2  =  8

B(x2, y2)  is B(7, 8)