Midpoint Formula

The midpoint formula is used to determine the point that is exactly halfway between two given points in a coordinate plane (the midpoint). Thus, a midpoint divides a line segment into 2 equal halves.

If A and B are two points with coordinates (x₁, y₁) and  (x₂, y₂), respectively, their midpoint can be found by the formula:

${\left( \dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\right)}$

Thus, the distance formula calculates the average of the given point’s x- and y-coordinates. 

Finding Midpoint Formula

Let us plot the points A (x₁, y₁) and B (x₂, y₂) on the coordinate plane. 

By joining the points ‘A’ and ‘B,’ we get a line segment AB. The midpoint of AB is the point halfway between A and B, so the point is M.

Here, the expression for the x-coordinate of the midpoint is ${\dfrac{x_{1}+x_{2}}{2}}$, which is the average of the x-coordinates. Similarly, the expression for the y-coordinate of the midpoint is ${\dfrac{y_{1}+y_{2}}{2}}$}}$, which is the average of the y-coordinates.

Thus, the midpoint is:

M = ${\left( \dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\right)}$

Find the midpoint of the two points, A (7, 9) and B (2, 5), using the midpoint formula.

Solution:

Given (x₁, y₁) = (7, 9), the coordinates of the first point
(x₂, y₂) = (2, 5), the coordinates of the second point
Now, applying the formula, we get
${\left( \dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\right)}$ 
= ${\left( \dfrac{7+2}{2},\dfrac{9+5}{2}\right)}$
= ${\left( \dfrac{9}{2},\dfrac{14}{2}\right)}$
= ${\left( \dfrac{9}{2},7\right)}$ or ${\left( 4.5,7\right)}$
Thus, the midpoint between the points A (7, 9) and B (2, 5) is ${\left( \dfrac{9}{2},7\right)}$ or ${\left( 4.5,7\right)}$

The endpoints of a line segment are (p, 5) and (6, 15). If the midpoint is (2, 10), find the value of p.

Solution:

Given (x₁, y₁) = (p, 5) and (x₂, y₂) = (6, 15)
As we know, the midpoint is
${\left( \dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\right)}$
Here, ${\left( \dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\right)}$ = (2, 10)
Comparing the x-coordinates,
${\dfrac{x_{1}+x_{2}}{2}}$ = 2
⇒ ${\dfrac{p+6}{2}}$ = 2
⇒ p = -2
Thus, the value of p is -2