Calculating C Using The Formula C=√(a²+b²) With A=24 And B=13
In mathematics, the Pythagorean Theorem is a fundamental concept that describes the relationship between the sides of a right triangle. It states that 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. This theorem is expressed by the formula: C = √(a² + b²), where C represents the hypotenuse, and a and b represent the lengths of the other two sides.
Understanding the Pythagorean Theorem and its Application
The Pythagorean Theorem is a cornerstone of geometry and trigonometry, providing a crucial link between the sides of a right triangle. To fully grasp its significance, let's delve deeper into its components and practical applications. In the formula C = √(a² + b²), C symbolizes the hypotenuse, the longest side of the right triangle, while a and b denote the lengths of the other two sides, often referred to as legs. The theorem's elegance lies in its ability to establish a direct relationship between these sides, allowing us to calculate the length of one side if the other two are known. Imagine a scenario where you're constructing a building and need to ensure that the walls form a perfect right angle. The Pythagorean Theorem comes to the rescue! By measuring the lengths of two walls (a and b) and applying the formula, you can determine the required length of the diagonal (hypotenuse C) to guarantee a precise 90-degree angle. This principle extends to various fields, including navigation, where sailors and pilots utilize the theorem to calculate distances and bearings. Architects and engineers rely on it to design stable structures, while carpenters employ it for precise woodworking and construction projects. The beauty of the Pythagorean Theorem lies not only in its mathematical elegance but also in its widespread applicability across diverse domains, making it an indispensable tool for problem-solving and real-world applications. Whether you're calculating the optimal path for a delivery route or designing a sturdy bridge, the Pythagorean Theorem serves as a reliable guide, ensuring accuracy and precision in your endeavors.
Applying the Formula to Solve for C
Now, let's apply this knowledge to a specific problem. We are given that a = 24 and b = 13, and our goal is to calculate C. This involves substituting the given values into the formula and performing the necessary calculations. Substituting the values of a and b into the formula C = √(a² + b²), we get C = √(24² + 13²). This equation sets the stage for our calculation, where we'll first square the individual values of a and b, then add the results, and finally take the square root to arrive at the value of C. It's a step-by-step process that showcases the power of the Pythagorean Theorem in determining the hypotenuse of a right triangle when the lengths of the other two sides are known. The beauty of this process lies in its simplicity and accuracy. By following the formula and substituting the given values, we can confidently calculate the length of the hypotenuse, a crucial parameter in various applications, from construction and engineering to navigation and design. The equation C = √(24² + 13²) serves as a bridge between the abstract world of mathematics and the concrete realities of the physical world, allowing us to solve real-world problems with precision and confidence. As we move forward with the calculation, each step will bring us closer to the final answer, reinforcing the theorem's practical significance and its ability to provide solutions in diverse scenarios.
Step-by-Step Calculation
The first step is to calculate the squares of a and b. We have a² = 24² = 576 and b² = 13² = 169. Now, we add these two values together: 576 + 169 = 745. The process of squaring a and b is fundamental to the Pythagorean Theorem, as it transforms linear measurements into areas, paving the way for calculating the hypotenuse. Squaring 24, we arrive at 576, representing the area of a square with sides of length 24. Similarly, squaring 13 yields 169, the area of a square with sides of length 13. These squares form the basis for understanding the relationship between the sides of the right triangle and the hypotenuse. The subsequent addition of 576 and 169, resulting in 745, represents the combined area of the squares formed by the legs of the triangle. This sum is crucial as it directly relates to the square of the hypotenuse, allowing us to determine its length. The calculation process is methodical and precise, ensuring that each step contributes to the final solution. The beauty of mathematics lies in its ability to break down complex problems into simpler steps, making them easier to understand and solve. This step-by-step approach not only provides a clear path to the answer but also reinforces the underlying principles of the Pythagorean Theorem, highlighting its practical significance in various fields. As we progress through the calculation, the connection between the formula and the numerical values becomes increasingly apparent, solidifying our understanding of the theorem's power and versatility.
Next, we need to find the square root of 745. Using a calculator, we find that √745 ≈ 27.29. Therefore, C ≈ 27.29. The act of finding the square root of 745 is the final step in unveiling the length of the hypotenuse, the missing piece of our right triangle puzzle. The square root operation essentially reverses the squaring process, allowing us to transition from the combined area of the squares formed by the legs to the linear measurement of the hypotenuse. Using a calculator, we obtain an approximate value of 27.29 for the square root of 745. This number represents the length of the hypotenuse, denoted as C, and completes our calculation. The approximation to two decimal places reflects the precision required in many practical applications, where accurate measurements are crucial for success. The process of finding the square root is not just a mathematical operation; it's a bridge that connects the abstract world of numbers to the tangible reality of physical lengths. This step highlights the power of mathematics to provide solutions that are both accurate and meaningful, allowing us to make informed decisions in various fields, from construction and engineering to navigation and design. The final value of C ≈ 27.29 represents the culmination of our calculations, demonstrating the elegance and practicality of the Pythagorean Theorem in solving real-world problems.
Conclusion
Therefore, given a = 24 and b = 13, we have calculated that C ≈ 27.29. This exercise demonstrates the practical application of the Pythagorean Theorem in finding the length of the hypotenuse of a right triangle. The journey from understanding the theorem to applying it to a specific problem showcases the power of mathematical reasoning and problem-solving. The Pythagorean Theorem, with its elegant formula C = √(a² + b²), serves as a cornerstone of geometry and trigonometry, providing a fundamental relationship between the sides of a right triangle. This relationship is not just a mathematical abstraction; it has real-world implications in various fields, from construction and engineering to navigation and design. The ability to calculate the length of the hypotenuse when the other two sides are known is a valuable skill, enabling us to solve problems related to distances, angles, and spatial relationships. The process of substituting the given values, performing the calculations, and arriving at the final answer reinforces the importance of precision and accuracy in mathematical work. The approximation to two decimal places reflects the practical needs of many applications, where precise measurements are essential for success. The final value of C ≈ 27.29 represents not just a number but a solution to a problem, a testament to the power of mathematical thinking and its ability to provide answers that are both meaningful and practical. As we continue to explore the world of mathematics, the Pythagorean Theorem will undoubtedly remain a valuable tool, guiding us through various challenges and helping us to make sense of the world around us.
