Coordinate Plane Formula

The Formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

When to use: Like a map with street numbers—the address (3, 2) is 3 right, 2 up.

Quick Example

Point (3, 4) is 3 right and 4 up from the origin; (-2, 1) is 2 left and 1 up.

Notation

(x, y) ordered pair, origin at (0, 0)

What This Formula Means

A two-dimensional surface formed by horizontal (x) and vertical (y) number lines.

Like a map with street numbers—the address (3, 2) is 3 right, 2 up.

Formal View

The Cartesian plane is the set \mathbb{R}^2 = \{(x, y) \mid x, y \in \mathbb{R}\} equipped with the Euclidean metric d((x_1,y_1),(x_2,y_2)) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

Worked Examples

Example 1

easy
In which quadrant is the point (-3, 5) located?

Solution

  1. 1
    The x-coordinate is -3 (negative, so left of the origin).
  2. 2
    The y-coordinate is 5 (positive, so above the origin).
  3. 3
    Negative x and positive y place the point in Quadrant II.

Answer

Quadrant II
The coordinate plane has four quadrants. The signs of the coordinates determine the quadrant: (+,+) is QI, (−,+) is QII, (−,−) is QIII, and (+,−) is QIV.

Example 2

medium
Find the distance between points (1, 2) and (4, 6).

Common Mistakes

  • Mixing up x and y order
  • Confusing quadrants

Why This Formula Matters

The coordinate plane is the foundation for graphing functions, analyzing geometry, and visualizing data patterns.

Frequently Asked Questions

What is the Coordinate Plane formula?

A two-dimensional surface formed by horizontal (x) and vertical (y) number lines.

How do you use the Coordinate Plane formula?

Like a map with street numbers—the address (3, 2) is 3 right, 2 up.

What do the symbols mean in the Coordinate Plane formula?

(x, y) ordered pair, origin at (0, 0)

Why is the Coordinate Plane formula important in Math?

The coordinate plane is the foundation for graphing functions, analyzing geometry, and visualizing data patterns.

What do students get wrong about Coordinate Plane?

x comes first (horizontal distance), then y (vertical distance) — this order is never reversed.

What should I learn before the Coordinate Plane formula?

Before studying the Coordinate Plane formula, you should understand: number sense, integers.

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