Formula To Find Midpoint, Definition with Examples

The midpoint formula in coordinate geometry helps us find the exact center point of a line segment. Instead of estimating or guessing the halfway point, this formula gives a precise method to calculate it using the coordinates of the line segment endpoints. This division always happens in a 1:1 ratio, meaning the midpoint lies exactly halfway along the line. Read this blog to understand the steps to find the midpoint of a line segment using the midpoint formula

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What is the Midpoint of a Line Segment?

A midpoint is defined as the point that is exactly halfway between two given points. This midpoint divides the line segment into two equal halves. Each half of the segment has the same length, and the midpoint lies on the same straight line connecting the two endpoints.

Mathematically, if a line segment has endpoints A(x1, y1) and B(x2, y2), the midpoint M is the average of the x-coordinates and the average of the y-coordinates of the two points.

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Definition of Midpoint Formula

The midpoint formula is a mathematical rule used to find the point that is exactly halfway between two given points in a coordinate plane.

It is defined as:

If A(x1, y1)  and B(x2, y2)are two points, then the midpoint M of the line segment AB is given by:

Midpoint (M)  =  (x1 + x2) /2,  (y1 + y2)/ 2

This point M divides the line segment AB into two equal parts.

What is the Formula of the Midpoint?

The formula of the midpoint is derived from the arithmetic mean of the coordinates of the endpoints. Here’s how it's presented:

Midpoint (M)  =  (x1 + x2) /2,  (y1 + y2)/ 2

This formula is universally applicable to all 2D Cartesian coordinates, whether they involve positive numbers, negative numbers, or decimals.

Let’s break it down:

• x-coordinate of M = the average of x1 and x2/2

• y-coordinate of M = the average of y1​ and y2/2

This simple calculation gives the precise centre point between two locations on a coordinate plane.

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Derivation of Midpoint Formula

Let’s consider a line segment AB with endpoints:

A(xA,yA)

B(xB,yB)

Let the coordinates of the midpoint be M(xM,yM)

We will now find expressions for xM,yM​ by considering how distances work on the coordinate axes.

On the X-axis:

• The x-coordinate of point A is xA

• The x-coordinate of point B is xB

Since the midpoint lies exactly halfway between these two points, its x-coordinate is the average of xA and xB​:

xM​ = xA + xB/2

On the Y-axis:

• The y-coordinate of point A is yA

• The y-coordinate of point B is yB

So, the y-coordinate of the midpoint is:

yM​ = yA + yB/2

Therefore, the midpoint M is given by:

Midpoint (M)  =  (xA + xB) /2,  (yA + yB)/ 2

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Real Life Examples of Midpoint Formula

The midpoint formula isn't limited to the classroom, it has real-life uses in various fields. Here are a few real-life examples where midpoint calculations are commonly applied:

Urban Planning and Construction: Midpoints help identify center locations for placing buildings, fountains, or other features equidistant from two points.

Computer Graphics: The midpoint is used in rendering and animation for calculating points between movements or designing curves and lines.

Navigation and GPS: Coordinates are averaged to estimate central meeting points between two people or vehicles.

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Midpoint Formula Solved Examples

Example 1:  Find the midpoint of the line joining points A(4,5) and B(6,7).

Solution:

 x = 4 + 6/ 2 = 10/ 2 = 5

y = 5 + 7/2 ​ = 12/2 ​=6

Midpoint = (5,6)

Example 2: Find the midpoint of the line joining A(6.4, 3) and B(−10.7, 4).

Solution:

x = 6.4 + (−10.7)/2 = −4.3/2 = −2.15

y = 3 + 4/2 ​= 7/2​ = 3.5

Midpoint = (−2.15, 3.5)

Example 3: If the midpoint is (1,0) and one endpoint is A(−6,−5), find the other endpoint B(x,y).

Solution:

Let B = (x,y)

Using the midpoint formula:

1 = −6 + x /2 ⇒ x = 8

0 = −5 + y ​⇒ y = 5

So, B = (8, 5)

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