The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
It measures the straight-line distance between two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$.
The formula is essential for problems involving geometric shapes like triangles, circles, and hyperbolas.
In three-dimensional space, the formula extends to $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.
Understanding how to derive the distance formula from the Pythagorean theorem can help in memorizing it.
Review Questions
What is the distance between points (3, 4) and (7, 1)?
How does the distance formula change when extended to three dimensions?
Derive the distance formula using the Pythagorean theorem.
Related terms
Coordinate Plane: A two-dimensional surface on which points are plotted and located by their coordinates $(x, y)$.
Pythagorean Theorem: A mathematical principle stating that in a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
Midpoint Formula: Calculates the midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ as $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.