2%2B(y2-y1)2.jpeg "(x2-x1)2+(y2-y1)2")
Understanding the Distance Formula: (x2-x1)2 + (y2-y1)2
The formula (x2 - x1)2 + (y2 - y1)2 is a fundamental concept in mathematics, representing the distance formula. This formula calculates the straight-line distance between two points in a two-dimensional coordinate plane.
Breaking Down the Formula
Let's break down the formula and understand its components:
- (x2 - x1)2: This part calculates the horizontal difference between the two points. It represents the square of the change in the x-coordinate.
- (y2 - y1)2: This part calculates the vertical difference between the two points. It represents the square of the change in the y-coordinate.
The square root of the sum of these squared differences gives us the distance between the two points.
Visualizing the Distance Formula
Imagine two points on a coordinate plane, labeled as (x1, y1) and (x2, y2). The distance formula essentially uses the Pythagorean theorem to find the length of the hypotenuse of a right triangle formed by these points.
- The horizontal distance (x2 - x1) forms one leg of the triangle.
- The vertical distance (y2 - y1) forms the other leg.
- The distance between the two points is the hypotenuse.
Applications of the Distance Formula
The distance formula has numerous applications in various fields, including:
- Geometry: Calculating the distance between points, finding the perimeter of shapes, and determining the properties of geometric figures.
- Physics: Calculating the magnitude of displacement, velocity, and acceleration in two-dimensional motion.
- Computer science: Developing algorithms for pathfinding, navigation, and image processing.
Example
Let's calculate the distance between the points (2, 3) and (5, 7) using the distance formula:
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Identify the coordinates:
- (x1, y1) = (2, 3)
- (x2, y2) = (5, 7)
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Apply the formula:
- Distance = √((5 - 2)2 + (7 - 3)2)
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Simplify:
- Distance = √(32 + 42)
- Distance = √(9 + 16)
- Distance = √25
- Distance = 5
Therefore, the distance between the points (2, 3) and (5, 7) is 5 units.
The distance formula is a versatile tool that allows us to measure distances in the two-dimensional coordinate plane, providing valuable insights in various fields of study.