The distance formula is a mathematical equation used to calculate the straight-line distance between two points in a coordinate plane. It is a fundamental concept in geometry and algebra that allows for the precise measurement of the length between any two points on a coordinate grid.
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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 the coordinate plane.
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.
The distance formula can be used to find the length of a line segment, the perimeter of a polygon, or the diameter of a circle on a coordinate plane.
The distance formula is an essential tool in solving problems related to geometric transformations, such as translations, reflections, and rotations, as well as in calculating the distance between two points in three-dimensional space.
Understanding the distance formula and its applications is crucial in solving a variety of problems in pre-algebra, algebra, and beyond, including those related to 9.7 Solve a Formula for a Specific Variable.
Review Questions
Explain how the distance formula is derived from the Pythagorean Theorem.
The distance formula is derived directly 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 two points can be viewed as the endpoints of a line segment that forms the hypotenuse of a right triangle. By applying the Pythagorean Theorem to this triangle, the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ is obtained, allowing for the calculation of the length of the line segment between the two points.
Describe how the distance formula can be used to solve problems related to 9.7 Solve a Formula for a Specific Variable.
In the context of 9.7 Solve a Formula for a Specific Variable, the distance formula can be used to rearrange and solve equations involving distances between points on a coordinate plane. For example, if a formula involves the distance between two points, the distance formula can be substituted to isolate a specific variable and solve for its value. This application of the distance formula is crucial in solving problems that require finding the value of a variable within a given formula, as it allows for the transformation of the equation into a form that can be more easily manipulated and solved.
Analyze how the distance formula can be used to determine the diameter of a circle on a coordinate plane.
The distance formula can be used to calculate the diameter of a circle on a coordinate plane by finding the distance between the two points that represent the endpoints of the circle's diameter. Specifically, if the coordinates of the center of the circle are $(x_1, y_1)$ and a point on the circle's circumference is $(x_2, y_2)$, then the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ can be used to find the length of the diameter, which is twice the radius of the circle. This application of the distance formula is important in solving problems related to the properties and measurements of circles on a coordinate plane.
A two-dimensional plane where points are identified by their x and y coordinates, allowing for the graphical representation of mathematical relationships.
A fundamental theorem in geometry that states the relationship between the lengths of the sides of a right triangle, which is the basis for the distance formula.
Midpoint Formula: A formula used to find the midpoint between two points on a coordinate plane, which is closely related to the distance formula.