Understanding the calculation of the length of a line segment in coordinate geometry is vital. This knowledge enables us to find the distance between two points on a Cartesian plane, a fundamental concept in many mathematical and real-world applications.

Introduction to the Distance Formula

The distance between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on a Cartesian plane is calculated using the distance formula, which is an application of Pythagoras' theorem:

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Calculating the Length of a Line Segment

Example 1: Basic Calculation

Find the length of the line segment between $A(1, 2)$ and $B(4, 6)$.

Solution:

1. Identify Coordinates:

$x_1 = 1$, $y_1 = 2$, $x_2 = 4$, $y_2 = 6$.

2. Apply the Formula:

The length of the line segment AB is 5 units.

Example 2: Negative Coordinates

Calculate the length of the line segment between $C(-3, -1)$ and $D(1, -5)$.

Solution:

1. Coordinates:

2. Apply the Formula:

The length of CD is $4\sqrt{2}$ units.

Practice Problems

Problem 1

Find the length of the line segment between $E(2, -3)$ and $F(-4, 1)$.

Solution:

1. Coordinates:

2. Apply the Formula:

Problem 2

Calculate the distance between $G(-2, -2)$ and $H(3, 3)$.

Solution:

1. Coordinates:

2. Apply the Formula:

Key Points

• Accuracy in Coordinates: Ensure coordinates are correctly placed within the formula.
• Handling Negative Values: Pay attention to negative signs to avoid calculation errors.
• Simplification: Simplify under the square root before calculating the final distance.
• Units of Measurement: Lengths are always positive and depend on the context for units.