Distance Formula

Definition

The distance formula is a mathematical equation used to calculate the distance between two points in a coordinate plane. It is a fundamental concept in geometry and algebra, with applications in various topics such as solving mixture and uniform motion problems, as well as understanding circles.

5 Must Know Facts For Your Next Test

1. The distance formula is derived from the Pythagorean Theorem and is used to calculate the length of the hypotenuse of a right triangle formed by the two points on a coordinate plane.
2. The distance formula is expressed as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.
3. The distance formula is used in the context of solving mixture and uniform motion applications, as the distance traveled is a key component in these types of problems.
4. In the topic of circles, the distance formula is used to determine the radius of a circle, as well as to find the distance between the center of a circle and a point on the circle's circumference.
5. The distance formula can also be used to find the length of a line segment on a coordinate plane, as well as to calculate the perimeter of a polygon.

Review Questions

• Explain how the distance formula is derived from the Pythagorean Theorem and how it is used to calculate the distance between two points on a coordinate plane.
  • The distance formula is derived from the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In the context of a coordinate plane, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ forms the hypotenuse of a right triangle, where the other two sides are the differences in the x-coordinates and the y-coordinates. By applying the Pythagorean Theorem, the distance formula is obtained as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This formula allows us to calculate the distance between any two points on a coordinate plane.
• Describe how the distance formula is used in the context of solving mixture and uniform motion applications, and explain its importance in these types of problems.
  • In the context of mixture and uniform motion applications, the distance formula is used to calculate the distance traveled by an object or the distance between two objects. For example, in a mixture problem, the distance formula can be used to determine the distance between two locations where different substances are mixed. In a uniform motion problem, the distance formula is used to calculate the distance traveled by an object given its initial position, final position, and the time elapsed. The distance formula is a crucial component in these types of problems because it allows you to quantify the spatial relationships between objects or locations, which is essential for setting up and solving the problem correctly.
• Analyze the role of the distance formula in the topic of circles, and explain how it is used to determine the radius of a circle and the distance between the center of a circle and a point on its circumference.
  • In the topic of circles, the distance formula plays a significant role. To determine the radius of a circle, the distance formula can be used to calculate the distance between the center of the circle and a point on the circle's circumference. This distance will be the radius of the circle. Additionally, the distance formula can be used to find the distance between the center of a circle and any point on the circle's circumference. This is important in understanding the properties and characteristics of circles, as the distance between the center and any point on the circumference should be equal to the radius of the circle. By applying the distance formula in the context of circles, you can gain a deeper understanding of the geometric properties and relationships within this fundamental shape.

Related terms

Coordinate Plane:

A two-dimensional plane with a horizontal x-axis and a vertical y-axis, used to represent and locate points in a two-dimensional space.

Pythagorean Theorem:

A fundamental theorem in geometry that states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Midpoint Formula:

A formula used to find the midpoint between two points on a coordinate plane, calculated as the average of the x-coordinates and the average of the y-coordinates.