Distance Between (-6, 2) And (8, 10) On A Coordinate Grid

Jul 12, 2025 by ADMIN 58 views

Calculating the Distance Between Two Points on a Coordinate Grid

In coordinate geometry, determining the distance between two points is a fundamental concept. This article delves into calculating the distance between the points $(-6, 2)$ and $(8, 10)$ on a coordinate grid. Understanding this calculation is essential for various mathematical and real-world applications, from navigation to computer graphics. We will explore the distance formula, its derivation, and a step-by-step application to solve the problem at hand. This comprehensive guide will not only provide the solution but also enhance your understanding of the underlying principles, ensuring you can confidently tackle similar problems in the future.

Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem, a cornerstone of Euclidean geometry. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse, and $a$ and $b$ are the other two sides.

To apply this to coordinate geometry, consider two points, $(x_1, y_1)$ and $(x_2, y_2)$ , on a coordinate plane. The horizontal distance between these points is $|x_2 - x_1|$ , and the vertical distance is $|y_2 - y_1|$ . These distances form the two legs of a right-angled triangle. The distance between the two points is the hypotenuse of this triangle. Therefore, using the Pythagorean theorem, we can derive the distance formula as:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This formula allows us to calculate the straight-line distance between any two points in a coordinate plane. The distance formula is a crucial tool in various fields, including geometry, trigonometry, and calculus. It forms the basis for many advanced concepts and applications. For instance, in computer graphics, it's used to calculate distances between objects in a virtual environment, and in navigation, it helps determine the shortest path between two locations. Understanding the distance formula is not just about memorizing a formula; it's about grasping the fundamental geometric principles that underlie it.

Applying the Distance Formula to Our Problem

Now, let's apply the distance formula to find the distance between the points $(-6, 2)$ and $(8, 10)$ . We'll follow a step-by-step approach to ensure clarity and accuracy.

Identify the Coordinates:

We have two points: $(-6, 2)$ and $(8, 10)$ . Let's designate $(-6, 2)$ as $(x_1, y_1)$ and $(8, 10)$ as $(x_2, y_2)$ . This assignment is arbitrary, and we would obtain the same result if we swapped the points. However, consistency in substitution is key to avoiding errors.
Substitute the Values into the Distance Formula:

The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . Substituting our coordinates, we get:

$d = \sqrt{(8 - (-6))^2 + (10 - 2)^2}$
Simplify the Expression Inside the Parentheses:

First, we simplify the terms inside the parentheses:

$8 - (-6) = 8 + 6 = 14$

$10 - 2 = 8$

So, our equation now looks like this:

$d = \sqrt{(14)^2 + (8)^2}$
Calculate the Squares:

Next, we square the numbers:

$14^2 = 196$

$8^2 = 64$

Our equation becomes:

$d = \sqrt{196 + 64}$
Add the Squared Values:

Now, we add the squared values together:

$196 + 64 = 260$

Our equation is now:

$d = \sqrt{260}$
Final Result:

Therefore, the distance between the points $(-6, 2)$ and $(8, 10)$ is $\sqrt{260}$ . This corresponds to option D in the given choices.

By following these steps, we have successfully calculated the distance between the two points using the distance formula. This methodical approach ensures accuracy and helps in understanding the underlying principles.

Analyzing the Options

Now that we have calculated the distance as $\sqrt{260}$ , let's examine the given options to confirm our answer and understand why the other options are incorrect.

A. $\sqrt{60}$ :

This option is significantly smaller than our calculated distance. It would arise from incorrect subtraction or squaring of the coordinate differences. For example, if we mistakenly calculated $(8 - (-6))$ as $2$ instead of $14$, or if we made errors in the squaring process, we might arrive at this answer. This option represents a substantial underestimation of the actual distance.

B. $\sqrt{68}$ :

This option is also smaller than $\sqrt{260}$, though closer than option A. It could result from errors in calculating the differences in the x and y coordinates or in the squaring and addition steps. This option indicates a partial understanding of the process but with significant calculation errors.

C. $\sqrt{162}$ :

This option is closer to the correct answer but still incorrect. It might stem from errors in the initial subtraction or in the squaring of the differences. For example, if there were a sign error or a miscalculation in squaring one of the terms, this result could arise. This option demonstrates a better grasp of the formula but still contains significant arithmetic mistakes.

D. $\sqrt{260}$ :

This is the correct answer, as we derived through our step-by-step calculation. It represents the accurate application of the distance formula and the correct execution of each step. This option confirms our understanding of the formula and our ability to apply it correctly.

By analyzing each option, we reinforce our understanding of the problem and the importance of accurate calculations. This analytical approach not only validates our answer but also enhances our problem-solving skills.

Common Mistakes to Avoid

When calculating the distance between two points, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accuracy in your calculations. Here are some of the most frequent errors:

Incorrect Subtraction of Coordinates:

A common mistake is subtracting the coordinates in the wrong order or making sign errors. For example, instead of calculating $(8 - (-6))$ , one might incorrectly calculate it as $(8 - 6)$ or $(-6 - 8)$ . This can lead to significant errors in the final result. Always double-check the order of subtraction and pay close attention to signs.
Sign Errors:

Sign errors are particularly problematic when dealing with negative coordinates. For instance, subtracting a negative number is equivalent to adding its positive counterpart. Misunderstanding this can lead to incorrect calculations. Careful attention to the rules of integer arithmetic is crucial.
Squaring Errors:

Another common mistake occurs when squaring the differences. For example, incorrectly squaring a number or making a sign error during squaring (e.g., squaring -14 and getting -196 instead of 196) can lead to a wrong answer. Always double-check your squares and remember that the square of any real number is non-negative.
Forgetting to Take the Square Root:

After calculating the sum of the squares, it's essential to take the square root to find the actual distance. Forgetting this final step will result in an answer that represents the square of the distance, not the distance itself. Always remember to take the square root at the end.
Mixing Up Coordinates:

It's crucial to maintain consistency when substituting the coordinates into the distance formula. Mixing up $x_1$ with $y_1$ or $x_2$ with $y_2$ can lead to incorrect calculations. Ensure you clearly identify and substitute the coordinates correctly.
Arithmetic Errors:

Simple arithmetic errors, such as mistakes in addition or subtraction, can also lead to incorrect results. Double-checking each step of the calculation can help catch these errors.

By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in calculating distances between points on a coordinate grid. Practice and careful attention to detail are key to mastering this skill.

Real-World Applications of Distance Calculation

The distance formula isn't just a theoretical concept; it has numerous practical applications in various fields. Understanding these applications can highlight the real-world relevance of this mathematical tool and enhance your appreciation for its importance.

Navigation:

In navigation, the distance formula is used extensively to calculate the distances between locations. GPS systems, for instance, rely on this formula to determine the distance between a user's current position and a destination. Whether it's mapping applications on smartphones or navigation systems in vehicles, the distance formula is a fundamental component.
Computer Graphics:

In computer graphics, calculating distances is essential for various tasks, such as rendering 3D scenes, collision detection, and animation. Determining the distance between objects in a virtual environment allows for realistic interactions and simulations. For example, in video games, the distance formula is used to check if two characters are close enough to interact or if a projectile has hit its target.
Geographic Information Systems (GIS):

GIS uses the distance formula to analyze spatial data, such as calculating the distance between cities, landmarks, or geographical features. This is crucial for urban planning, environmental studies, and resource management. For instance, GIS can be used to determine the optimal location for a new facility based on its distance from existing infrastructure and population centers.
Robotics:

In robotics, calculating distances is vital for robot navigation, path planning, and obstacle avoidance. Robots need to be able to accurately determine their distance from objects in their environment to move safely and efficiently. The distance formula enables robots to map their surroundings and plan optimal routes.
Physics and Engineering:

In physics and engineering, the distance formula is used in various calculations, such as determining the trajectory of projectiles, analyzing the motion of objects, and designing structures. For example, engineers use the distance formula to calculate the length of cables needed for a bridge or the distance a projectile will travel when launched at a certain angle.
Sports Analytics:

In sports analytics, the distance formula can be used to track player movements, measure distances covered during a game, and analyze player performance. This information can be used to develop strategies, improve training techniques, and enhance player evaluations. For instance, in soccer, the distance formula can help track how far a player has run during a match.
Real Estate:

In real estate, the distance formula can be used to calculate the distance between properties, amenities, and points of interest. This can help potential buyers or renters assess the convenience of a location. For example, calculating the distance to schools, parks, or public transportation can be valuable in making informed decisions.

The wide range of applications demonstrates the versatility and importance of the distance formula. By understanding its real-world relevance, you can better appreciate its significance and motivate your learning.

Conclusion

In conclusion, the distance between the points $(-6, 2)$ and $(8, 10)$ on a coordinate grid is $\sqrt{260}$ . We arrived at this answer by applying the distance formula, which is derived from the Pythagorean theorem. This formula is a fundamental tool in coordinate geometry and has numerous applications in various fields, including navigation, computer graphics, and engineering.

We walked through a step-by-step calculation, highlighting the importance of accurate substitution, simplification, and arithmetic. We also analyzed the given options, understanding why the incorrect options were not valid. Furthermore, we discussed common mistakes to avoid, such as incorrect subtraction, sign errors, and forgetting to take the square root.

By understanding the distance formula, its derivation, and its applications, you can confidently solve similar problems and appreciate the real-world relevance of this mathematical concept. Practice and attention to detail are key to mastering this skill. This article has provided a comprehensive guide to calculating the distance between two points, equipping you with the knowledge and skills to tackle such problems effectively.