Midpoint Formula Revision Notes for Leaving Cert Mathematics

Midpoint Formula

What is the Midpoint Formula?

The midpoint formula provides a way to find the exact centre point of a line segment defined by two points in a Cartesian plane. If the two points are $(x_1, y_1)$ and $(x_2, y_2)$ , the midpoint $M$ is calculated as:

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

This formula averages the $x$ -coordinates and the $y$ -coordinates of the endpoints of the segment.

Why Use the Midpoint Formula?

The midpoint formula is used to:

Divide a line segment into two equal parts.
Identify the centre of a geometric figure.
Solve real-world problems like finding the middle of a journey.

How is the Formula Derived?

The midpoint is the average of the $x$ -coordinates and the $y$ -coordinates of the two points because it is equidistant from both endpoints of the segment.

For example, if $x_1$ and $x_2$ are the $x$ -coordinates of two points, the midpoint's $x$ -coordinate is $\frac{x_1 + x_2}{2}$ . The same logic applies to the $y$ -coordinates.

Worked Examples

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Example 1: Find the Midpoint of a Segment

Problem: Find the midpoint of the segment joining $A(2, 3)$ and $B(4, 7)$ .

Solution:

Substitute $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (4, 7)$ into the formula:

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

M = \left( \frac{2 + 4}{2}, \frac{3 + 7}{2} \right) = \left( \frac{6}{2}, \frac{10}{2} \right) = (3, 5)

Answer: The midpoint is (3, 5)

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Example 2: Determine the Centre of a Diagonal

Problem: A rectangle has a diagonal connecting $(-3, -2)$ and $(5, 6)$ .

Find the centre of the diagonal.

Solution:

Substitute $(x_1, y_1) = (-3, -2)$ and $(x_2, y_2) = (5, 6)$

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

M = \left( \frac{-3 + 5}{2}, \frac{-2 + 6}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2)

Answer: The centre of the diagonal is (1, 2)

Summary

The midpoint formula calculates the exact centre of a line segment:

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

To use the formula, average the $x$ -coordinates and $y$ -coordinates of the two endpoints.
Applications include geometry, coordinate problems, and real-life scenarios.
Practice applying the formula to gain confidence in its use.

Midpoint Formula (Leaving Cert Mathematics): Revision Notes