The distance between two points on a coordinate grid can be calculated using the Pythagorean theorem. So, what is the distance between the points (-6, 2) and (8, 10) on a coordinate grid? Let’s find out.

To determine the distance, we need to calculate the difference in x-coordinates and y-coordinates of the two points. In this case, the difference in x-coordinates is 8 – (-6) = 14, and the difference in y-coordinates is 10 – 2 = 8.

Applying the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite to the right angle) is equal to the sum of squares of the other two sides. In our case, these “sides” represent differences in x and y coordinates.

## What Is The Distance Between (–6, 2) And (8, 10) On A Coordinate Grid?

### Understanding Coordinate Grids

Coordinate grids are essential tools used in mathematics and geometry to represent points and locations in a two-dimensional plane. They provide a visual representation of the relationships between different points on a grid, allowing us to analyze distances, angles, and patterns. In a coordinate grid, each point is identified by an ordered pair (x, y), where x represents the horizontal distance and y represents the vertical distance from a reference point.

The horizontal line on the grid is called the x-axis, while the vertical line is known as the y-axis. The point where these two lines intersect is called the origin, denoted as (0, 0). From this point, we can navigate through different quadrants of the coordinate plane: Quadrant I in the upper right, Quadrant II in the upper left; Quadrant III in the lower left; and Quadrant IV in the lower right.

### Calculating Distance on a Coordinate Grid

One fundamental concept in coordinate grids is calculating distances between two points. To determine the distance between any two points (x1, y1) and (x2, y2), including (-6, 2) and (8, 10) mentioned earlier, we use a formula known as “the distance formula.” This formula enables us to find straight-line or Euclidean distances on a coordinate plane.

The distance formula states that d = √((x2 – x1)^2 + (y2 – y1)^2), where d represents distance. Applying this formula to our example coordinates yields:

d = √((8 – (-6))^2 + (10 – 2)^2) = √(14^2 + 8^2) ≈ √(196 + 64) ≈ √260 ≈ 16.12

Therefore, using this formula, the distance between (-6, 2) and (8, 10) on a coordinate grid is approximately 16.12 units.

### Applying The Distance Formula

The distance formula is not only limited to finding distances between two points in a coordinate grid but also has various applications across different fields. For instance:

- Navigation: The distance formula helps determine the shortest route between two locations on a map or GPS system.
- Physics: It enables physicists to calculate displacements and distances traveled by objects in motion.
- Computer Graphics: The distance formula allows programmers to render realistic images by determining the proximity of objects in virtual environments.
- Surveying: Surveyors use this formula to measure distances between points on land accurately.

By understanding and applying the distance formula, we can explore and analyze relationships between points on a coordinate grid, providing us with valuable insights across multiple disciplines.

Remember that when using the distance formula, it’s crucial to pay attention to signs (+/-) while subtracting coordinates and ensure accurate calculations for precise results.

## Applying The Distance Formula to (–6, 2) and (8, 10)

To find the distance between two points on a coordinate grid, we can use the distance formula. In this case, we want to determine the distance between point A at (-6, 2) and point B at (8, 10). Let’s dive into how we can apply the distance formula to calculate this distance.

The distance formula is derived from the Pythagorean theorem and helps us find the straight-line distance between any two points in a Cartesian plane. It is expressed as:

d = √((x₂ – x₁)² + (y₂ – y₁)²)

Where d represents the distance between the two points, (x₁, y₁) are the coordinates of one point, and (x₂, y₂) are the coordinates of another point.

Now let’s plug in our values for point A (-6, 2) and point B (8, 10):

d = √((8 – (-6))² + (10 – 2)²) = √(14² + 8²) = √(196 + 64) = √260 ≈ 16.12

Therefore, using the distance formula applied to points A (-6, 2) and B (8 ,10), we find that they are approximately 16.12 units apart on a coordinate grid.

Remember that knowing how to apply formulas correctly is essential for accurate calculations. So next time you encounter a similar scenario, you’ll be well-equipped to find the distance between any two points on a coordinate grid using the distance formula.