What is the fundamental geometric theorem that provides the basis for the distance formula?

Pythagoras' Theorem ($a^2 + b^2 = c^2$). The horizontal and vertical changes between points form the legs of a right triangle, and the distance is the hypotenuse.

How does the distance formula differ from the gradient formula in terms of coordinate usage?

The distance formula squares the differences and sums them under a square root, while the gradient formula calculates the ratio of the vertical difference to the horizontal difference.

Why is the result of the distance formula always a non-negative number?

Because the differences are squared, which produces non-negative results, and the principal square root of a non-negative sum is always non-negative.

What is a common error when calculating the distance between points with negative coordinates, such as $(-2, -3)$ and $(4, -5)$?

Incorrectly handling the subtraction of negatives (e.g., thinking $-3 - (-5)$ is $-8$ instead of $2$). Using brackets like $(-3) - (-5)$ helps prevent this.

Does the order in which you subtract the coordinates (e.g., $x_1 - x_2$ vs $x_2 - x_1$) change the final distance?

No. Because the difference is squared, the result is the same. For example, $(5 - 2)^2 = 3^2 = 9$ and $(2 - 5)^2 = (-3)^2 = 9$.

What is the specific formula for the distance $d$ between points $(x_1, y_1)$ and $(x_2, y_2)$?

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This calculates the square root of the sum of the squared horizontal and vertical displacements.

When should you leave your answer in surd form (e.g., $\sqrt{13}$) rather than a decimal?

You should use surd form when an 'exact' answer is required. Decimals are usually approximations unless the square root results in a terminating decimal.

What happens to the distance calculation if the line segment is perfectly horizontal?

The vertical difference $(y_2 - y_1)$ becomes zero. The formula simplifies to $\sqrt{(x_2 - x_1)^2}$, which is just the absolute difference between the x-coordinates.

What is the 'Triangle Inequality' check for a line segment's length?

The calculated distance (hypotenuse) must be shorter than the sum of the horizontal and vertical legs, but longer than either leg individually.

How do you calculate the distance of a point $(x, y)$ from the origin $(0, 0)$?

The formula simplifies to $d = \sqrt{x^2 + y^2}$, as the differences from zero are simply the coordinates themselves.

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Length of a Line

Summary

The length of a line segment in a 2D Cartesian plane is the straight-line distance between two points. It is calculated by applying Pythagoras' Theorem to the horizontal and vertical displacements between the coordinates.

1. Definition & Core Concepts

The length of a line segment represents the absolute distance between two distinct points, $(x_1, y_1)$ and $(x_2, y_2)$ , on a coordinate grid.
This distance is a scalar quantity, meaning it only has magnitude and is always non-negative, regardless of the direction of the line.
The calculation relies on finding the horizontal change (the 'run') and the vertical change (the 'rise') between the two endpoints.

A coordinate geometry diagram showing a line segment as the hypotenuse of a right-angled triangle, with horizontal and vertical legs representing the differences in x and y coordinates.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions