The midpoint exercises of a line segment can be solved by applying the midpoint formula. This formula is derived considering that the coordinates of the midpoint are found by adding the respective coordinates of the points and dividing by 2.
Here, we will look at the midpoint formula and apply it to solve some practice problems.
Summary of the formula for the midpoint of a line segment
We can calculate the coordinates of the midpoint of a line segment considering that the x-coordinates of the midpoint are equal to half the sum of the x-coordinates of the two points that define the segment.
Similarly, the y-coordinates of the midpoint are equal to half the sum of the y-coordinates of the two points. Therefore, if we have points A and B with the coordinates $latex A = (x_{1}, y_{1})$ and $latex B = (x_{2}, y_{2})$, the formula the midpoint is:
| Formula for the midpoint $latex M=\left(\frac{x_{1}+x_{2}}{2}+\frac{y_{1}+y_{2}}{2}\right)$ |
The midpoint will be expressed as the coordinates $latex M = (x_{3}, y_{3})$.
Midpoint of a line segment – Examples with answers
Each of the following examples has its respective solution. However, try to solve the problems yourself by applying the formula for the midpoint of a segment.
EXAMPLE 1
What is the midpoint between the points (2, 6) and (8, 12)?
Solution
We have the following coordinates:
- $latex (x_{1}, y_{1})=(2, 6)$
- $latex (x_{2}, y_{2})=(8, 12)$
We use the coordinates given in the midpoint formula:
$latex M=\left(\frac{x_{1}+x{2}}{2},\frac{y_{1}+y{2}}{2}\right)$
$latex =\left(\frac{2+8}{2},\frac{6+12}{2}\right)$
$latex =\left(\frac{10}{2},\frac{18}{2}\right)$
$latex =(5, 9)$
The coordinates of the midpoint are $latex M = (5, 9)$.
EXAMPLE 2
Determine the midpoint between the points (5, 7) and (9, 13).
Solution
We have the following points:
- $latex (x_{1}, y_{1})=(5, 7)$
- $latex (x_{2}, y_{2})=(9, 13)$
We apply the midpoint formula with these points:
$latex M=\left(\frac{x_{1}+x{2}}{2},\frac{y_{1}+y{2}}{2}\right)$
$latex =\left(\frac{5+9}{2},\frac{7+13}{2}\right)$
$latex =\left(\frac{14}{2},\frac{20}{2}\right)$
$latex =(7,10)$
The coordinates of the midpoint are $latex M=(7, 10)$.
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EXAMPLE 3
Find the midpoint if we have the points (-5, -6) and (6, -2).
Solution
We write the coordinates as follows:
- $latex (x_{1}, y_{1})=(-5, -6)$
- $latex (x_{2}, y_{2})=(6, -2)$
In this case, we have negative coordinates, but we simply use the midpoint formula as in the previous exercises:
$latex M=\left(\frac{x_{1}+x{2}}{2},\frac{y_{1}+y{2}}{2}\right)$
$latex =\left(\frac{-5+6}{2},\frac{-6-2}{2}\right)$
$latex =\left(\frac{1}{2},\frac{-8}{2}\right)$
$latex =\left(\frac{1}{2}, -4\right)$
The midpoint has the coordinates $latex M=\left(\frac{1}{2}, -4\right)$.
EXAMPLE 4
We have the points (p, -2) and (8, 10). Find the value of p if the midpoint is (2, 4).
Solution
We have the following information:
- $latex (x_{1}, y_{1})=(p, -2)$
- $latex (x_{2}, y_{2})=(8, 10)$
- $latex (x_{3}, y_{3})=(2, 4)$
By applying the midpoint formula with the given information, we have:
$latex M=\left(\frac{x_{1}+x{2}}{2},\frac{y_{1}+y{2}}{2}\right)$
$latex =\left(\frac{p+8}{2},\frac{-2+10}{2}\right)$
In this case, we have to find the value of p, which is part of the x coordinates of the midpoint. Therefore, we form an equation considering that the given midpoint has an x coordinate of 2:
$latex 2=\left(\frac{p+8}{2}\right)$
$latex 2=p+8$
$latex p=-6$
The value of p is -6.
EXAMPLE 5
If the diameter of a circle is defined by the points (-4, 2) and (2, 8), what are the coordinates of the center of the circle?
Solution
We have the following values:
- $latex (x_{1}, y_{1})=(-4,2)$
- $latex (x_{2}, y_{2})=(2,8)$
The center of the circle is located exactly in the middle of the diameter, so we find the midpoint of the two given points:
$latex M=\left(\frac{x_{1}+x{2}}{2},\frac{y_{1}+y{2}}{2}\right)$
$latex =\left(\frac{-4+2}{2},\frac{2+8}{2}\right)$
$latex =\left(\frac{-2}{2},\frac{12}{2}\right)$
$latex =(-1,7)$
The center of the circle is located at $latex (-1, 7)$.
Midpoint of a line segment – Practice problems
Use the midpoint formula with the given information to solve the following problems. If you need help with this, you can look at the solved examples above.
See also
Interested in learning more about distance, midpoint, and slope on the plane? Take a look at these pages:
