Distance Between (0,-2) & (-9,-10): Simplified Solution

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Finding the Distance Between Two Points: A Comprehensive Guide

Hey guys! Let's dive into a common math problem: finding the distance between two points. This might seem tricky at first, but with the right formula and a bit of practice, you'll be solving these like a pro. In this article, we'll specifically tackle the problem of finding the distance between the points (0, -2) and (-9, -10), ensuring our answer is either a whole number or a fully simplified radical expression, without any rounding. So, let's get started!

Understanding the Distance Formula

The distance formula is the key to solving these problems. It's derived from the Pythagorean theorem, which you might remember from geometry. The formula allows us to calculate the straight-line distance between any two points in a coordinate plane. To truly grasp how to calculate the distance between two points, it's essential to break down the distance formula and understand its components. The distance formula is mathematically expressed as:

The Formula

(x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Where:

  • (x1,y1)(x_1, y_1) are the coordinates of the first point.
  • (x2,y2)(x_2, y_2) are the coordinates of the second point.

This formula might look intimidating, but don't worry! It's just a matter of plugging in the values and following the steps. The brilliance of the distance formula lies in its foundation within the Pythagorean theorem, a cornerstone of geometry. This connection not only provides a method for calculation but also offers a deeper understanding of spatial relationships in a coordinate plane.

Breaking Down the Formula

Think of it this way: We're essentially creating a right triangle where the distance between the two points is the hypotenuse. The legs of the triangle are the differences in the x-coordinates and the y-coordinates. The distance formula helps us apply the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the length of that hypotenuse (which is the distance we're looking for!). To effectively utilize the distance formula, one must first recognize its structural components and understand their geometrical significance. The terms (x2−x1)(x_2 - x_1) and (y2−y1)(y_2 - y_1) represent the horizontal and vertical distances between the points, respectively. These differences form the legs of a right triangle, where the distance between the two points is the hypotenuse. Squaring these differences, as the formula dictates, allows us to apply the Pythagorean theorem, a fundamental concept in Euclidean geometry that relates the sides of a right triangle.

Applying the Distance Formula to Our Problem

Now, let's apply this formula to our specific problem. We have two points: (0, -2) and (-9, -10). Let's identify our values:

  • x1=0x_1 = 0
  • y1=−2y_1 = -2
  • x2=−9x_2 = -9
  • y2=−10y_2 = -10

Plugging in the Values

Next, we'll substitute these values into the distance formula: To ensure accuracy and clarity in applying the distance formula, a systematic approach to substituting the given values is crucial. This process involves carefully replacing each variable in the formula with its corresponding numerical value, which helps in minimizing errors and facilitates a clearer understanding of the computational steps. Before plunging into calculations, it's often beneficial to visually represent the points on a coordinate plane. This visual aid not only helps in conceptualizing the problem but also serves as a quick check for the reasonableness of the answer. For instance, plotting the points (0, -2) and (-9, -10) can give a rough estimate of the distance, which can then be compared against the calculated result.

(−9−0)2+(−10−(−2))2\sqrt{(-9 - 0)^2 + (-10 - (-2))^2}

Simplifying the Expression

Now, let's simplify step by step: Simplifying mathematical expressions efficiently requires a methodical approach, especially when dealing with square roots and exponents. The key is to break down the problem into smaller, manageable steps, ensuring each operation is performed correctly. Following the order of operations (PEMDAS/BODMAS) is crucial in this process. This involves first addressing the operations within parentheses, then exponents, followed by multiplication and division, and finally addition and subtraction. When dealing with square roots, it's important to simplify the expression inside the square root as much as possible before attempting to find the root itself. This may involve combining like terms, performing arithmetic operations, and simplifying any fractions or radicals that appear within the expression. A systematic approach not only minimizes errors but also makes the process more transparent and easier to follow.

(−9)2+(−8)2\sqrt{(-9)^2 + (-8)^2}

81+64\sqrt{81 + 64}

145\sqrt{145}

Checking for Simplification

Our result is 145\sqrt{145}. Now, we need to check if we can simplify this radical further. This involves looking for perfect square factors within 145. When simplifying radicals, the primary goal is to extract any perfect square factors from the radicand (the number under the square root symbol). This process not only simplifies the expression but also presents the answer in its most concise and mathematically elegant form. To achieve this, one must identify the factors of the radicand and determine if any of them are perfect squares (i.e., numbers that can be expressed as the square of an integer). For instance, perfect squares include numbers like 4, 9, 16, 25, and so on. Once a perfect square factor is identified, it can be factored out of the radicand, and its square root can be taken and placed outside the radical symbol. This technique is particularly useful in algebra and calculus, where simplified radical expressions are often required for further calculations and manipulations.

Prime Factorization of 145

Let's find the prime factors of 145:

145 is divisible by 5: 145 / 5 = 29

So, the prime factors of 145 are 5 and 29. Neither of these are perfect squares, so 145\sqrt{145} is already in its simplest form.

The Final Answer

Therefore, the distance between the points (0, -2) and (-9, -10) is 145\sqrt{145}. And that's our final answer, guys! We've successfully found the distance and expressed it in its simplest radical form. When tackling mathematical problems, especially those involving formulas and calculations, adopting a systematic and meticulous approach can significantly improve accuracy and understanding. This approach typically begins with a clear restatement of the problem, followed by the identification of relevant formulas and principles that can be applied to solve it. Each step in the problem-solving process should be clearly articulated and justified, ensuring that the logic behind the solution is transparent and easy to follow. It's also crucial to pay close attention to detail, particularly when substituting numerical values into formulas and performing arithmetic operations.

Tips for Solving Distance Problems

Before we wrap up, here are a few tips to keep in mind when solving distance problems:

  • Double-check your values: Make sure you've correctly identified x1x_1, y1y_1, x2x_2, and y2y_2.
  • Be careful with signs: Pay close attention to negative signs, especially when subtracting.
  • Simplify completely: Always check if you can simplify the radical further.
  • Visualize the points: If you're having trouble, try plotting the points on a graph. This can give you a visual sense of the distance.

Practice Makes Perfect

The best way to master the distance formula is through practice. Try working through different examples with varying coordinates. The more you practice, the more comfortable you'll become with the formula and the process. Regular practice is an indispensable component of mastering any mathematical concept, particularly those that involve formulas and procedures. Engaging in consistent practice sessions not only reinforces the understanding of the underlying principles but also hones the skills necessary to apply these principles effectively in a variety of contexts. For instance, when learning the distance formula, working through a multitude of examples with varying coordinates helps solidify the ability to accurately substitute values and perform the calculations required to find the distance between two points.

Conclusion

Finding the distance between two points doesn't have to be daunting. By understanding the distance formula and practicing regularly, you can tackle these problems with confidence. Remember to break down the problem into steps, double-check your work, and always simplify your answer as much as possible. Keep practicing, and you'll become a distance-finding whiz in no time! So keep up the awesome work, and I'll catch you in the next math adventure!