Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points on a coordinate plane. This formula calculates the average of the \( x \)- and \( y \)-coordinates of two points to determine their midpoint. It is commonly used in geometry and other applications that involve dividing line segments or finding center points.

Formula for the Midpoint between Two Points

If we have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) between these points is given by the formula:

\( M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \)

In this formula:

  • \( x_1 \) and \( y_1 \) are the \( x \)- and \( y \)-coordinates of the first point
  • \( x_2 \) and \( y_2 \) are the \( x \)- and \( y \)-coordinates of the second point
  • \( M \) represents the midpoint coordinates, which are the average of the corresponding coordinates of the two points

Detailed Explanation of the Formula

The midpoint formula \( M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \) works by calculating the average of the \( x \)-coordinates and the \( y \)-coordinates of two points. This process gives the location of the midpoint by averaging the horizontal and vertical positions between the two points.

  • Add the \( x \)-coordinates of both points and divide by 2 to get the \( x \)-coordinate of the midpoint.
  • Add the \( y \)-coordinates of both points and divide by 2 to get the \( y \)-coordinate of the midpoint.

Example 1: Finding the Midpoint between Two Points

Problem: Find the midpoint of the points \( (2, 3) \) and \( (8, 7) \) on a coordinate plane.

Solution:

  1. Write down the formula: \( M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \).
  2. Substitute \( x_1 = 2 \), \( y_1 = 3 \), \( x_2 = 8 \), and \( y_2 = 7 \): \( M = \left( \dfrac{2 + 8}{2}, \dfrac{3 + 7}{2} \right) \).
  3. Add the \( x \)- and \( y \)-coordinates: \( M = \left( \dfrac{10}{2}, \dfrac{10}{2} \right) \).
  4. Divide by 2: \( M = (5, 5) \).

The midpoint of the points \( (2, 3) \) and \( (8, 7) \) is \( (5, 5) \).

Example 2: Using the Midpoint Formula in Real-Life Context

Problem: A fence is being constructed to divide a plot of land between two markers located at \( (4, 2) \) and \( (-6, 8) \). Find the midpoint where the fence should start.

Solution:

  1. Write down the formula: \( M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \).
  2. Substitute \( x_1 = 4 \), \( y_1 = 2 \), \( x_2 = -6 \), and \( y_2 = 8 \): \( M = \left( \dfrac{4 + (-6)}{2}, \dfrac{2 + 8}{2} \right) \).
  3. Add the \( x \)- and \( y \)-coordinates: \( M = \left( \dfrac{-2}{2}, \dfrac{10}{2} \right) \).
  4. Divide by 2: \( M = (-1, 5) \).

The midpoint where the fence should start is at \( (-1, 5) \).

The midpoint formula is useful in geometry, design, and engineering for finding center points, dividing line segments, and more.