Math Problem Statement
please explain the distance formula
Solution
The distance formula is used to find the distance between two points in a coordinate plane. It's derived from the Pythagorean Theorem and works in both two-dimensional and three-dimensional spaces.
Formula in Two Dimensions (2D)
If you have two points, and , the distance between these points is given by:
Explanation
- Difference in Coordinates: The formula calculates the horizontal distance and the vertical distance between the two points.
- Squaring: Squaring these differences removes any negative signs and gives a measure of distance squared.
- Summation: Adding the squared distances gives the total squared distance between the points.
- Square Root: Finally, taking the square root of the sum converts it back into the actual distance.
Example Calculation
Consider points and . Using the formula:
Formula in Three Dimensions (3D)
If the points have three coordinates, such as and , the formula is extended as:
Would you like more examples, details, or have any specific questions?
Related Questions:
- How is the distance formula derived from the Pythagorean Theorem?
- How does the distance formula work in real-life applications?
- How do you use the distance formula to determine if points are collinear?
- What is the difference between the 2D and 3D distance formulas?
- Can the distance formula be used with complex numbers?
Tip:
Always double-check the subtraction steps when using the distance formula to ensure that the signs are correct.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Plane
Formulas
Distance formula in 2D: d = √((x2 - x1)^2 + (y2 - y1)^2)
Distance formula in 3D: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 8-10
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