A midpoint is a coordinate point that is halfway between two other points on a line segment. Midpoints occur in both two dimensional space on a graph and three dimensional space inside of a cube, sphere, or other shapes. Finding the midpoint helps calculate geographical, computer programing, and economic problems in the real world.

The midpoint formula in coordinate geometry is an equation that calculates the halfway point distance between two known coordinate points. The midpoint formula in coordinate geometry is also used to find the coordinates of the endpoint if we know the coordinates of the other endpoint and the midpoint.

What is Midpoint?

The midpoint is the point that lies exactly halfway between two other points in space. In mathematics, it can be defined as the point that is equidistant from the endpoints of a line segment or a continuous curve. 

Consider any two points, say A and C, the midpoint is a point B which is located halfway between points A and C. Observe that point B is equidistant from A and C from the below image.

Important Point: A midpoint exists only for a line segment. A line or a ray cannot have a midpoint because a line is indefinite in both directions and a ray has only one end and thus can be extended.

Midpoint Formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the coordinate geometry midpoint formula.

Suppose \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) be the endpoints of a line segment. The midpoint of a line formula is equal to half of the sum of the \(x\)-coordinates of the two points, and half of the sum of the \(y\)-coordinates of the two points. 

So, the midpoint is the point whose \(x\)-coordinate is equal to the average of the \(x\)-coordinates of the two endpoints and whose \(y\)-coordinate is equal to the average of the \(y\)-coordinates of the two endpoints. The mathematical form of the midpoint of a line formula is given under the below header.

To learn more about the midpoint theorem and midpoint theorem formula, please click here.

Midpoint Formula in Math

In a two-dimensional coordinate system, the midpoint \((x_m, y_m)\) of a line segment connecting the points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) can be found using the formula:

\((x_m, y_m)\) = \(\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\).

Let us understand this with the help of an example. Find the midpoint of two points in a one-dimensional axis. In a one-dimensional axis, to calculate the midpoint, we can simply measure the length of the line segment and divide by 2.

Suppose, we have two points, \(5\) and \(9\), on a number line. The midpoint will be calculated as: 

\(\frac{(5+9)}{2}=\frac{14}{2}=7\)

So, \(7\) is the midpoint of \(5\) and \(9\) as shown in the image.

Derivation of Midpoint Formula

Follow the below steps to derive the midpoint formula:

Step 1: Consider a line segment with its endpoints, \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\). Plot these points as shown in the below image. 

Step 2: Now, we know for any line segment, the midpoint is halfway between its two endpoints. 

Step 3: So, the expression for the \(x\)-coordinate of the midpoint is \(\frac{(x_{1}+x_{2})}{2}\), which is the average of the \(x\)-coordinates of the two endpoints. 

Step 4: Similarly, we can get the expression for the \(y\)-coordinate is \(\frac{(y_{1}+y_{2})}{2}\), which is the average of the \(y\)-coordinates of the two endpoints.

Step 5: Therefore, the midpoint of a line segment formula is \(\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\).

How to Find Midpoint? 

There are two ways or methods to find the midpoint of the line joining the two points which is based on the points and their coordinate values and these are given below.

Method 1: Midpoint on a Horizontal Line

If the line segment is vertical or horizontal, then dividing the length by 2 and counting that value from any of the endpoints, this will give the midpoint of the line segment. 

Let us understand this with the help of an example. Let the coordinates of points A and B are \((-3, 2)\) and \((1, 2)\) as shown in the below midpoint formula graph. Here, the length of horizontal line AB is \(4\) units. When we divide \(4\) by \(2\), we get \(2\) units. Moving \(2\) units from the point \((-3, 2)\) will give \((-1, 2)\). So, \((-1, 2)\) is the midpoint of AB.

Method 2: Coordinate Representation of Midpoint

The second method to find the midpoint is using the midpoint formula. For example, let the coordinates of points A and B are \((-3, -3)\) and \((1, 4)\) as shown below in the midpoint formula graph. Now using the midpoint formula, we have: \(\left(\frac{-3+1}{2},\frac{-3+4}{2}\right)=\left(\frac{-2}{2},\frac{1}{2}\right)=\left(-1,\frac{1}{2}\right)\). So, \(\left(-1,\frac{1}{2}\right) \) is the midpoint of AB. 

Midpoint Formula Uses

• The midpoint formula is often used to find the midpoint in order to bisect a line segment. It can also be used to prove that a line segment is bisected.
• The midpoint formula is used to find the average, or median of two values. This formula is often utilized with maps and with the distance formula.

Formulas Related to Midpoint

The midpoint formula in coordinate geometry is the computation separately for the \(x\)-coordinate of the points, and the \(y\)-coordinate of the points. On the other hand, the computations of points between two given points also include a similar computation of the \(x\)-coordinate and the \(y\)-coordinate of the given points. The following two formulas are closely related to the midpoint formula and these are:

• Centroid of triangle formula
• Section formula

Centroid of a Triangle Formula

The centroid of a triangle is the centre of the triangle. It is referred to as the point of concurrency of medians of a triangle.

For a triangle with vertices \((x_{1}, y_{1})\), \((x_{2}, y_{2})\), and \((x_{3}, y_{3})\) the formula to find the coordinates of the centroid of the triangle is as follows:

Midpoint of a triangle formula = \(\left(\frac{(x_{1}+x_{2}+x_{3})}{2},\frac{(y_{1}+y_{2}+y_{3})}{2}\right)\).

Section Formula

The section formula is used to find the coordinates of the point that divides a line segment (externally or internally) into some ratio.

Consider we have a point \(P(x, y)\) that divides the line segment with marked points as \(A(x_{1}, y_{1})\) and \(B(x_{2}, y_{2})\). To find the coordinates, we use the section midpoint formula, which is mathematically expressed as: 

\(\left(\frac{(mx_{2} \pm nx_{1})}{m \pm n},\frac{(my_{2} \pm ny_{1})}{m \pm n}\right)\). 

Midpoint Formula Summary

• The midpoint is the point that lies exactly halfway between two other points in space.
• The midpoint formula, in geometry, is an equation that calculates the halfway point distance between two known coordinate points.
• In a one-dimensional axis, to calculate the midpoint, we can simply measure the length of the line segment and divide by 2.
• In a two-dimensional coordinate system, the midpoint of a line segment connecting the points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) can be found using the formula \(\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\).
• The midpoint divides a line segment in an equal ratio, that is, \(1:1\). 

Note: The midpoint formula class 10th topic is very important for the students of class 10th.

Midpoint Formula Solved Examples 

1.Find the midpoint of a line whose endpoints are \((4, 5)\) and \((6, 7)\).

Solution: Let \((x_{1}, y_{1}) = (4, 5)\), and \((x_{2}, y_{2}) = (6, 7)\).

According to the formula we can find the midpoint \((x, y)\):

\((x,y)=\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\)

\(\Rightarrow\) \((x,y)=\left(\frac{(4+6)}{2},\frac{(5+7)}{2}\right)\)

\(\Rightarrow\) \((x,y)=\left(\frac{(10)}{2},\frac{(12)}{2}\right)\) 

\(\Rightarrow\) \((x,y)=\left(5,6\right)\) 

Therefore, the midpoint of a line whose endpoints are \((4, 5)\) and \((6, 7)\) is \((5, 6)\).

2.The endpoints of a line segment are \((2, h)\) and \((4, 7)\). Find the value of \(h\) if the midpoint is \((3, -1)\).

Solution: Let \((x_{1}, y_{1}) = (2, h)\), and \((x_{2}, y_{2}) = (4, 7)\).

According to the definition of midpoint we have, \(\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)=\left(\frac{(2+4)}{2},\frac{(h+7)}{2}\right)=\left(3,\frac{(h+7)}{2}\right)\) 

Equalizing this with the midpoint value \((3, -1)\), we have

\(\frac{(h+7)}{2} = -1\)

\(\Rightarrow\) \(h + 7 = -2\)

\(\Rightarrow\) \(h = -2 - 7\)

\(\Rightarrow\) \(h = -9\) 

Therefore, the value of \(h\) is \(-9\).

3.Find the coordinates of the centre of the circle whose endpoints of a diameter are \((0, 2)\), \((3, 4)\).

Solution: Given coordinates of endpoints of a diameter are:

\((0, 2) = (x_{1}, y_{1})\), and

\((3, 4) = (x_{2}, y_{2})\).

Centre of the circle is the midpoint of diameter.

Coordinates of the centre of a circle \(= \left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\)

\(\Rightarrow\) \(= \left(\frac{(0+3)}{2},\frac{(2+4)}{2}\right)\) 

\(\Rightarrow\) \(= \left(\frac{3}{2},\frac{6}{2}\right)\) 

\(\Rightarrow\) \(= \left(1.5,3\right)\) 

So, the coordinates of the centre of the circle whose endpoints of a diameter are \((0, 2)\), \((3, 4)\) is \(\left(1.5,3\right)\). 

4.If \((1, 0)\) is the midpoint of the line joining the points A\((-6, -5)\) and B, then find the coordinates of B.

Solution: Given, \((1, 0)\) is the midpoint of A and B.

A \(= (-6, -5) = (x_{1}, y_{1})\).

Let the coordinates of B are \((a, b)\), i.e. \((a, b) = (x_{2}, y_{2})\)

Using the midpoint formula,

\((x,y)=\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\Rightarrow (1,0)=\left(\frac{a-6}{2},\frac{b-5}{2}\right)\)

Now equating the \(x\) and \(y\) coordinates,

\(\frac{a-6}{2}=1, \frac{b-5}{2}=0 \)

\(a - 6 = 2, b - 5 = 0\)

\(a = 8, b = 5\)

Therefore, the coordinates of B is \((8, 5)\).

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