Distance formula

Definition

The distance formula calculates the distance between two points in a Cartesian coordinate system. It is derived from the Pythagorean theorem and expressed as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

5 Must Know Facts For Your Next Test

1. The distance formula is crucial for determining the straight-line distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.
2. It is derived from the Pythagorean theorem applied to a right triangle formed by the two points.
3. To use the formula, subtract the x-coordinates and y-coordinates of the points, square these differences, sum them up, and take the square root.
4. If points are given in three dimensions, it extends to $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.
5. This formula is extensively used in analytic geometry to find distances within geometric shapes like circles and polygons.

Review Questions

• What is the distance between the points (3, 4) and (6, 8)?
• How does the distance formula relate to the Pythagorean theorem?
• Extend the distance formula to three-dimensional space.

Related terms

Pythagorean Theorem:

A fundamental relation in Euclidean geometry among the three sides of a right triangle: $a^2 + b^2 = c^2$.

Cartesian Coordinate System: A coordinate system that specifies each point uniquely by a pair of numerical coordinates based on perpendicular axes.

Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ provides the midpoint of a line segment connecting two points.