(x2-x1)2+(y2-y1)2

Jasonbradley

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Jun 17, 2024

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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:

1. Identify the coordinates:

   • (x1, y1) = (2, 3)
   • (x2, y2) = (5, 7)

2. Apply the formula:

   • Distance = √((5 - 2)2 + (7 - 3)2)

3. 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.