Distance Formula in the Coordinate Plane

Distance in the Coordinate Plane

Distance Formula

The distance formula is used to find the length between two points in the xy-plane. It is derived from the Pythagorean Theorem and is fundamental in College Algebra and analytic geometry.

• Definition: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

• Application: This formula calculates the straight-line (Euclidean) distance between any two points in the plane.

Example Calculation

Find the distance between the points $(-6, 1)$ and $(0, -17)$.

• Step 1: Identify coordinates: $(x_1, y_1) = (-6, 1)$ and $(x_2, y_2) = (0, -17)$.

• Step 2: Substitute into the formula:

$d = \sqrt{(0 - (-6))^2 + (-17 - 1)^2}$

$= \sqrt{(6)^2 + (-18)^2}$

$= \sqrt{36 + 324}$

$= \sqrt{360}$

$\approx 18.97$

• Final Answer: The distance between $(-6, 1)$ and $(0, -17)$ is approximately 18.97 units (rounded to the nearest hundredth).

Additional info:

• The distance formula is a direct application of the Pythagorean Theorem in the context of coordinate geometry.

• This concept is foundational for later topics such as equations of circles, midpoint formula, and analytic proofs.