Calculate Distance Between Points (-15.4, -5.7) & (5.1, 14.2)

Hey there, math enthusiasts! Today, we're diving into a fundamental concept in coordinate geometry: calculating the distance between two points. This is a crucial skill in various fields, from navigation to computer graphics, and even everyday problem-solving. So, let's break it down and make sure you've got a solid grasp on it. We'll specifically tackle the problem of finding the distance between the points (-15.4, -5.7) and (5.1, 14.2), rounding our answer to the nearest tenth. Get ready to sharpen those pencils (or keyboards!) and let's get started!

Understanding the Distance Formula

Before we jump into the specific problem, let's refresh our understanding of the distance formula. This formula is the cornerstone of our calculations, derived directly from the Pythagorean theorem. Remember that good old friend from geometry class? The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²). Think of the two points we're given as defining the endpoints of the hypotenuse of a right triangle. The legs of this triangle are parallel to the x and y axes, and their lengths correspond to the differences in the x-coordinates and y-coordinates of the two points. This is the very foundation of how we find the distance between two points on a coordinate plane. The distance formula is essentially an application of the Pythagorean Theorem, making it a powerful tool for solving a wide variety of geometric problems. So, take a moment to really visualize this relationship between the points, the triangle, and the formula – it will make the whole process much clearer! Imagine a coordinate plane, picture two points plotted on it, and then visualize the right triangle that connects them. The distance between the points is simply the length of the longest side (the hypotenuse) of that triangle. Got it? Great! Now, let's translate that visual understanding into the actual formula and see how it works in practice. Understanding the origin and logic behind the distance formula will not only help you remember it better but also allow you to apply it confidently in various situations. If you ever forget the formula itself, you can always fall back on the fundamental principles of the Pythagorean Theorem and re-derive it. That's the beauty of understanding the underlying concepts – you're not just memorizing, you're truly learning!

The distance formula is expressed as follows:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

• (x₁, y₁) are the coordinates of the first point.
• (x₂, y₂) are the coordinates of the second point.

In simpler terms, guys, we find the difference in the x-coordinates, square it, find the difference in the y-coordinates, square that, add those two squares together, and then take the square root. Easy peasy, right?

Applying the Distance Formula to Our Points

Now, let's put this formula into action with our specific points: (-15.4, -5.7) and (5.1, 14.2). First, we need to identify which point is (x₁, y₁) and which is (x₂, y₂). It actually doesn't matter which one we choose as the first point, as long as we're consistent throughout the calculation. For the sake of clarity, let's assign:

• (x₁, y₁) = (-15.4, -5.7)
• (x₂, y₂) = (5.1, 14.2)

Now we simply substitute these values into the distance formula:

Distance = √[(5.1 - (-15.4))² + (14.2 - (-5.7))²]

See how we're carefully plugging in the x and y coordinates into their respective places in the formula? This is a crucial step, and accuracy here is key to getting the correct answer. Make sure you double-check your substitutions to avoid any silly mistakes! It's like following a recipe – if you miss an ingredient or misplace the quantities, the final dish won't turn out quite right. Similarly, in math, even a small error in substitution can lead to a significantly different result. So, take your time, be meticulous, and ensure that each value is placed correctly. Once you've mastered the art of substitution, the rest of the calculation becomes much smoother and more straightforward. Think of it as building a strong foundation – a solid start ensures a stable and accurate result. And remember, practice makes perfect! The more you work with the distance formula, the more confident and efficient you'll become at substituting values and solving problems. Don't be afraid to tackle different examples and challenge yourself – it's the best way to truly understand the concept and its application. So, let's keep going, carefully substituting the values and moving one step closer to finding the distance between our two points!

Step-by-Step Calculation

Let's break down the calculation step-by-step to minimize errors. First, we'll focus on the expressions inside the parentheses:

1. Calculate the difference in x-coordinates: 5. 1 - (-15.4) = 5.1 + 15.4 = 20.5
2. Calculate the difference in y-coordinates: 6. 2 - (-5.7) = 14.2 + 5.7 = 19.9

Now, we square these differences:

1. Square the difference in x-coordinates: 7. 5² = 420.25
2. Square the difference in y-coordinates: 8. 9² = 396.01

Next, we add the squared differences:

420.25 + 396.01 = 816.26

Finally, we take the square root of the sum:

Distance = √816.26 ≈ 28.57

Rounding to the Nearest Tenth

The problem asks us to round the decimal to the nearest tenth. Looking at our result, 28.57, the digit in the tenths place is 5, and the digit in the hundredths place is 7. Since 7 is greater than or equal to 5, we round up the 5 to a 6. Therefore, the distance rounded to the nearest tenth is 28.6.

Remember the rules for rounding, y'all? It's a simple but essential skill in mathematics and everyday life. When rounding to the nearest tenth, we look at the digit in the hundredths place. If it's 5 or greater, we round up the tenths digit. If it's less than 5, we leave the tenths digit as it is. This ensures that we're providing the most accurate approximation possible while adhering to the specified level of precision. Rounding might seem like a minor detail, but it can significantly impact the final answer, especially in practical applications where precision is crucial. Imagine, for example, calculating the dimensions of a room for flooring or the dosage of medication – even small rounding errors can lead to significant discrepancies. So, always pay close attention to the instructions regarding rounding and apply the rules consistently. In our case, rounding 28.57 to the nearest tenth gives us 28.6, which is the final answer to our distance problem. We've successfully navigated through all the steps, from understanding the distance formula to performing the calculations and rounding the result. Give yourselves a pat on the back for making it this far! Now, let's take a moment to recap the key takeaways and reinforce our understanding of the concepts involved.

Conclusion

So, the distance between the points (-15.4, -5.7) and (5.1, 14.2), rounded to the nearest tenth, is approximately 28.6 units. We've successfully applied the distance formula, navigated the calculations, and rounded our answer appropriately. This exercise demonstrates the power of the distance formula in determining the separation between two points in a coordinate plane. Remember, the distance formula is a versatile tool that has applications far beyond textbook problems. It's used in computer graphics to calculate distances between objects, in navigation systems to determine the shortest routes, and in many other fields where spatial relationships are important. By mastering this fundamental concept, you're not just learning a formula; you're developing a valuable skill that will serve you well in various aspects of your academic and professional life. Keep practicing, exploring different examples, and challenging yourself to apply the distance formula in new and creative ways. The more you work with it, the more comfortable and confident you'll become in your ability to solve distance-related problems. And who knows, you might even discover some exciting new applications of the distance formula along the way! So, keep exploring, keep learning, and keep pushing your mathematical boundaries – the possibilities are endless!

Remember the key takeaways:

• The distance formula is derived from the Pythagorean theorem.
• Careful substitution of values is crucial for accuracy.
• Rounding to the correct decimal place is important for the final answer.

Keep practicing, and you'll become a distance-calculating pro in no time!