Midpoint Calculator
A midpoint is a point on a line that is equidistant from both endpoints of the line or the arc.
To find the midpoint in a line, measure the distance between the two endpoints of a line and divide it by two. The result should be the mid-point you are looking for.
Midpoint Formula
The midpoint \( M \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Example: Suppose the distance of a line is 14cm, dividing the length by two gives us 7cm. Therefore the mid-point should be 7cm from the start.
Suppose coordinates of a line are given. (2,9) and (8,17).
How to find the midpoint:
It’s quite simple. We just need to add the value of the x-axis and divide by 2. In the same way, we need to add the value of the y-axis and divide it by two. The mid-point is (2+8/2,9+17/2), (5,13).
Midpoint Word Problems
Problem 1
Find the midpoint of the line segment connecting the points \( A(2, 3) \) and \( B(6, 7) \).
Solution: The midpoint \( M \) is given by:
\[ M = \left( \frac{2 + 6}{2}, \frac{3 + 7}{2} \right) = \left( \frac{8}{2}, \frac{10}{2} \right) = (4, 5) \]
Problem 2
The endpoints of a line segment are \( A(-1, 4) \) and \( B(3, 2) \). What is the midpoint?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-1 + 3}{2}, \frac{4 + 2}{2} \right) = \left( \frac{2}{2}, \frac{6}{2} \right) = (1, 3) \]
Problem 3
Find the midpoint between the points \( A(0, 0) \) and \( B(10, 10) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{0 + 10}{2}, \frac{0 + 10}{2} \right) = (5, 5) \]
Problem 4
Given the points \( A(-3, -2) \) and \( B(1, 4) \), find the midpoint.
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-3 + 1}{2}, \frac{-2 + 4}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1) \]
Problem 5
The endpoints of a line segment are \( A(5, 10) \) and \( B(15, 20) \). What is the midpoint?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{5 + 15}{2}, \frac{10 + 20}{2} \right) = \left( \frac{20}{2}, \frac{30}{2} \right) = (10, 15) \]
Problem 6
Find the midpoint of the line segment connecting \( A(-4, 0) \) and \( B(2, 8) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-4 + 2}{2}, \frac{0 + 8}{2} \right) = \left( \frac{-2}{2}, \frac{8}{2} \right) = (-1, 4) \]
Problem 7
Calculate the midpoint between the points \( A(1, 1) \) and \( B(7, 5) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{1 + 7}{2}, \frac{1 + 5}{2} \right) = \left( \frac{8}{2}, \frac{6}{2} \right) = (4, 3) \]
Problem 8
Given the points \( A(6, 12) \) and \( B(14, 4) \), find the midpoint.
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{6 + 14}{2}, \frac{12 + 4}{2} \right) = \left( \frac{20}{2}, \frac{16}{2} \right) = (10, 8) \]
Problem 9
Find the midpoint of the line segment connecting \( A(3, 8) \) and \( B(-5, 2) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{3 - 5}{2}, \frac{8 + 2}{2} \right) = \left( \frac{-2}{2}, \frac{10}{2} \right) = (-1, 5) \]
Problem 10
The endpoints of a line segment are \( A(0, 5) \) and \( B(4, 9) \). What is the midpoint?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{0 + 4}{2}, \frac{5 + 9}{2} \right) = \left( \frac{4}{2}, \frac{14}{2} \right) = (2, 7) \]
Problem 11
The points \( A(2, -3) \) and \( B(6, 1) \) are the endpoints of a line segment. What is the midpoint?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{2 + 6}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1) \]
Problem 12
Find the midpoint of the segment connecting the points \( A(-2, 5) \) and \( B(4, -1) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-2 + 4}{2}, \frac{5 - 1}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2) \]
Problem 13
Calculate the midpoint of the line segment joining \( A(8, 4) \) and \( B(10, 12) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{8 + 10}{2}, \frac{4 + 12}{2} \right) = \left( \frac{18}{2}, \frac{16}{2} \right) = (9, 8) \]
Problem 14
Given the points \( A(1, 7) \) and \( B(9, 3) \), find the midpoint of the line segment.
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{1 + 9}{2}, \frac{7 + 3}{2} \right) = \left( \frac{10}{2}, \frac{10}{2} \right) = (5, 5) \]
Problem 15
Find the midpoint between \( A(12, -6) \) and \( B(6, 0) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{12 + 6}{2}, \frac{-6 + 0}{2} \right) = \left( \frac{18}{2}, \frac{-6}{2} \right) = (9, -3) \]
Problem 16
Calculate the midpoint of the segment connecting \( A(0, 0) \) and \( B(20, 30) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{0 + 20}{2}, \frac{0 + 30}{2} \right) = (10, 15) \]
Problem 17
Given points \( A(-5, 5) \) and \( B(5, -5) \), find the midpoint.
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-5 + 5}{2}, \frac{5 - 5}{2} \right) = \left( \frac{0}{2}, \frac{0}{2} \right) = (0, 0) \]
Problem 18
Find the midpoint of the line segment connecting the points \( A(3, 3) \) and \( B(9, 15) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{3 + 9}{2}, \frac{3 + 15}{2} \right) = \left( \frac{12}{2}, \frac{18}{2} \right) = (6, 9) \]
Problem 19
The endpoints of a line segment are \( A(10, 2) \) and \( B(2, 8) \). What is the midpoint?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{10 + 2}{2}, \frac{2 + 8}{2} \right) = \left( \frac{12}{2}, \frac{10}{2} \right) = (6, 5) \]
Problem 20
Find the midpoint of the segment joining the points \( A(-6, 4) \) and \( B(4, -2) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-6 + 4}{2}, \frac{4 - 2}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1) \]
Problem 21
The endpoints of a line segment are \( A(3, -4) \) and \( B(7, 2) \). What is the midpoint of this segment?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{3 + 7}{2}, \frac{-4 + 2}{2} \right) = \left( \frac{10}{2}, \frac{-2}{2} \right) = (5, -1) \]
Problem 22
Find the midpoint of the segment connecting points \( A(-1, 5) \) and \( B(5, 11) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-1 + 5}{2}, \frac{5 + 11}{2} \right) = \left( \frac{4}{2}, \frac{16}{2} \right) = (2, 8) \]
Problem 23
The coordinates of point \( A \) are \( (2, 3) \) and the coordinates of point \( B \) are \( (10, 7) \). Determine the midpoint of the line segment \( AB \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{2 + 10}{2}, \frac{3 + 7}{2} \right) = \left( \frac{12}{2}, \frac{10}{2} \right) = (6, 5) \]
Problem 24
Calculate the midpoint of the line segment connecting points \( A(1, 2) \) and \( B(-3, -6) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{1 - 3}{2}, \frac{2 - 6}{2} \right) = \left( \frac{-2}{2}, \frac{-4}{2} \right) = (-1, -2) \]
Problem 25
The points \( A(0, 0) \) and \( B(8, 8) \) are the endpoints of a line segment. Find the midpoint.
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{0 + 8}{2}, \frac{0 + 8}{2} \right) = \left( \frac{8}{2}, \frac{8}{2} \right) = (4, 4) \]
Problem 26
Given points \( A(10, 0) \) and \( B(14, 12) \), calculate the midpoint of the segment.
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{10 + 14}{2}, \frac{0 + 12}{2} \right) = \left( \frac{24}{2}, \frac{12}{2} \right) = (12, 6) \]
Problem 27
The points \( A(-7, 5) \) and \( B(1, -3) \) define a line segment. What is the midpoint of this segment?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-7 + 1}{2}, \frac{5 - 3}{2} \right) = \left( \frac{-6}{2}, \frac{2}{2} \right) = (-3, 1) \]
Problem 28
Calculate the midpoint of the line segment joining points \( A(-2, -1) \) and \( B(4, 5) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{-2 + 4}{2}, \frac{-1 + 5}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2) \]
Problem 29
Find the midpoint of the segment connecting points \( A(6, 8) \) and \( B(10, 14) \).
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{6 + 10}{2}, \frac{8 + 14}{2} \right) = \left( \frac{16}{2}, \frac{22}{2} \right) = (8, 11) \]
Problem 30
The points \( A(4, 2) \) and \( B(-2, 6) \) define a line segment. What is the midpoint of this segment?
Solution: The midpoint \( M \) is:
\[ M = \left( \frac{4 - 2}{2}, \frac{2 + 6}{2} \right) = \left( \frac{2}{2}, \frac{8}{2} \right) = (1, 4) \]
