Table of Contents

**Introduction:**

In mathematics, a midpoint is crucial when dealing with various geometric figures and objects. It allows us to find the exact center between two given points, providing valuable information for calculations and analysis.

**Understanding the Midpoint**

Before diving into the formula and derivation, let’s grasp the concept of a midpoint. In geometry, the midpoint refers to the exact center of a line segment. It is equidistant from both ends of the segment, dividing it into two equal parts. The coordinates of the midpoint can be determined by using a simple formula.

**The formula for Finding the Midpoint**

To find the midpoint between two points, (x₁, y₁) and (x₂, y₂), we use the following formula:

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

**Derivation of the Midpoint Formula**

To derive the midpoint formula, we consider the coordinates of two points, A and B, which define a line segment. Let’s assume that A has coordinates (x₁, y₁) and B has coordinates (x₂, y₂).

To find the midpoint M, we notice that it lies exactly halfway between A and B. Therefore, the x-coordinate of M should be equidistant from the x-coordinates of A and B, while the y-coordinate of M should be equidistant from the y-coordinates of A and B.

Now, consider the x-coordinate of the midpoint. To find it, we add the x-coordinates of A and B and divide the sum by 2. Mathematically, it can be represented as:

x-coordinate of M = (x₁ + x₂) / 2

Similarly, for the y-coordinate, we add the y-coordinates of A and B and divide the sum by 2, resulting in:

y-coordinate of M = (y₁ + y₂) / 2

By combining these two equations, we obtain the midpoint formula:

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

**Example**

Example 1: Finding the Midpoint of Two Points

Let’s consider two points, A(2, 4) and B(6, 10). To find the midpoint, we will use the midpoint formula:

Midpoint = ((2 + 6) / 2, (4 + 10) / 2)

= (8 / 2, 14 / 2)

= (4, 7)

Therefore, the midpoint between A(2, 4) and B(6, 10) is (4, 7).

Midpoint Formula in Three-Dimensional Space:

The midpoint formula can also be extended to three-dimensional space, where we consider the x, y, and z coordinates. The formula for finding the midpoint between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is as follows:

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)

Example 2: Finding the Midpoint in Three-Dimensional Space

Let’s find the midpoint between two points, A(1, 2, 3) and B(4, 6, 8). Using the midpoint formula, we have:

Midpoint = ((1 + 4) / 2, (2 + 6) / 2, (3 + 8) / 2)

= (5 / 2, 8 / 2, 11 / 2)

= (2.5, 4, 5.5)

Hence, the midpoint between A(1, 2, 3) and B(4, 6, 8) is approximately (2.5, 4, 5.5).

Also, read Supplementary angles

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