Table of Contents
Last modified on August 30th, 2024
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.
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.
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.
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
Last modified on August 30th, 2024