Distance formula

Definition

The distance formula calculates the distance between two points in a plane. It is derived from the Pythagorean theorem.

5 Must Know Facts For Your Next Test

1. The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
2. It measures the straight-line distance between two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$.
3. The formula is essential for problems involving geometric shapes like triangles, circles, and hyperbolas.
4. In three-dimensional space, the formula extends to $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.
5. 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 $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.