Midpoint Formula

The midpoint formula is used to find a point (its coordinate values) that is located exactly between two other points in a plane. The formula finds its tremendous application in geometry.

The coordinates of the point (x, y) that lies exactly halfway between the two points (x1, y1) and (x2, y2) are given by:

` x = ( x_(1 )+ x_2)/2 , y = ( y_(1 )+ y_2)/2 `

Similarly, if we want to find the midpoint of a segment in the 3-dimensional space, we can determine the midpoint using:


` x = ( x_(1 )+ x_2)/2 , y = ( y_(1 )+ y_2)/2 , z = ( z_(1 )+ z_2)/2 `

The figure shown below gives an illustration of the midpoint formula.

Midpoint Formula
Explanation:

While calculating the midpoint between two sets of coordinates of a line segment, if we assume, point A is x1, y1 and x2, y2. Using the above midpoint formula, the average of the x coordinates and y coordinates give the midpoint of Point A and Point B. It is possible to find the midpoint of a vertical, horizontal and even a diagonal line segment using this formula.

Midpoint Formula Examples



Example 1:

Explanation:

Using the coordinate values as `x_1 = -1, y_1 = 4, x_2 = 3, y_2 = 6`

The coordinates of the Midpoint can be found using the midpoint formula:


` x = ( x_(1 )+ x_2)/2 , y = ( y_(1 )+ y_2)/2 `


` => x = ( -1+ 3 )/2 , y = ( 4 + 6 )/2 `


` => x = 1 , y = 5 `

Therefore, the midpoint is (1,5).

We can prove that the point (1,5) lies halfway between the two given points using distance formula discussed in the previous section.


Example 2:

Explanation:

Let us assume that the coordinates of point B are (x', y'). Now by making use of the midpoint formula:


` x = ( x_(1 )+ x_2)/2 , y = ( y_(1 )+ y_2)/2 `


Here we have `x_1 = 7, y_1 = -10, x_2 = x', y_2 = y', x = 4, y = -6`


` => 4 = ( 7 + x^')/2 , -6 = (-10 + y^')/2 `

Multiplying on both sides by 2 gives


` => 8 = 7 + x' , -12 = -10 + y' `


` => x' = 1 , y' = -2 `

Therefore, the midpoint is (1,-2).






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