Midpoint of a Line Segment Revision Notes for Leaving Cert Mathematics

Midpoint of a Line Segment

What is a midpoint?

The midpoint of a line segment is the point that lies exactly halfway between two endpoints. When you have a line segment connecting two points, the midpoint divides this segment into two equal parts of the same length.

Think of it like finding the centre point between two locations - it's the spot that is equally distant from both ends.

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Understanding the concept of a midpoint is fundamental to coordinate geometry. The key insight is that a midpoint creates two segments of identical length on either side.

In the diagram above, point M is the midpoint of line segment AB. You can see that M sits exactly in the middle, with equal distances from A to M and from M to B.

The midpoint formula

When you know the coordinates of two endpoints, you can find the midpoint using this formula:

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For a line segment joining points A(x₁, y₁) and B(x₂, y₂), the midpoint M has coordinates:

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

This formula tells us that:

The x-coordinate of the midpoint is the average of the two x-coordinates
The y-coordinate of the midpoint is the average of the two y-coordinates

Step-by-step method

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Finding the Midpoint - Systematic Approach

To find the midpoint of any line segment, follow these steps:

Identify the coordinates of both endpoints
Add the x-coordinates of both points and divide the result by 2
Add the y-coordinates of both points and divide the result by 2
Write your answer as coordinates (x, y)

Worked example 1

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Worked Example: Finding the Midpoint of A(-1, 3) and B(5, 7)

Step 1: Identify the coordinates

Point A: x₁ = -1, y₁ = 3
Point B: x₂ = 5, y₂ = 7

Step 2: Apply the midpoint formula

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

$= \left(\frac{-1 + 5}{2}, \frac{3 + 7}{2}\right)$

$= \left(\frac{4}{2}, \frac{10}{2}\right)$

$= (2, 5)$

Therefore, the midpoint is (2, 5).

Worked example 2

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Worked Example: Finding the Midpoint of C(0, -2) and D(6, 4)

Step 1: Identify the coordinates

Point C: x₁ = 0, y₁ = -2
Point D: x₂ = 6, y₂ = 4

Step 2: Apply the midpoint formula

$\text{Midpoint} = \left(\frac{0 + 6}{2}, \frac{-2 + 4}{2}\right)$

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

$= (3, 1)$

Therefore, the midpoint is (3, 1).

Worked example 3

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Worked Example: Finding the Midpoint of P(-3, -1) and Q(1, -5)

Step 1: Identify the coordinates

Point P: x₁ = -3, y₁ = -1
Point Q: x₂ = 1, y₂ = -5

Step 2: Apply the midpoint formula

$\text{Midpoint} = \left(\frac{-3 + 1}{2}, \frac{-1 + (-5)}{2}\right)$

$= \left(\frac{-2}{2}, \frac{-6}{2}\right)$

$= (-1, -3)$

Therefore, the midpoint is (-1, -3).

Exam tips

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Common Mistakes to Avoid

Double-check your arithmetic when adding coordinates and dividing by 2
Be careful with negative numbers - remember that adding a negative is the same as subtracting
Always write your final answer as coordinates in the form (x, y)
Show your working clearly - examiners like to see the midpoint formula written out

Remember!

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Key Points to Remember:

The midpoint is the point exactly halfway between two endpoints
Use the formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Add the x-coordinates and divide by 2 to get the x-coordinate of the midpoint
Add the y-coordinates and divide by 2 to get the y-coordinate of the midpoint
Always double-check your arithmetic, especially with negative numbers

Midpoint of a Line Segment (Leaving Cert Mathematics): Revision Notes