Calculating C Using Pythagorean Theorem With A=24 And B=13

Introduction

In this article, we will delve into the application of the Pythagorean theorem to calculate the length of the hypotenuse (C) of a right triangle. The Pythagorean theorem is a fundamental concept in geometry that relates 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 can be represented by the formula: C = √(a² + b²), where 'a' and 'b' are the lengths of the other two sides, and 'C' is the length of the hypotenuse. We will apply this formula using the given values of a = 24 and b = 13, rounding the answer to two decimal places as necessary. This exercise not only reinforces understanding of the theorem but also highlights its practical application in various fields, including construction, navigation, and engineering.

Understanding the Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of Euclidean geometry, providing a fundamental relationship between the three sides of a right triangle. A right triangle, by definition, is a triangle that contains one angle of 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are often referred to as legs or cathetus. The theorem, named after the ancient Greek mathematician Pythagoras, allows us to calculate the length of any one side of a right triangle if we know the lengths of the other two sides. This principle is not just a theoretical concept; it has numerous real-world applications, ranging from simple tasks like calculating the length of a ladder needed to reach a certain height to more complex problems in fields like architecture, surveying, and even computer graphics. The beauty of the theorem lies in its simplicity and its broad applicability. By understanding and applying the Pythagorean Theorem, we can solve a multitude of geometric problems and gain a deeper appreciation for the mathematical relationships that govern the world around us.

Applying the Formula with Given Values

To calculate the length of the hypotenuse (C) using the formula C = √(a² + b²), we are given the values a = 24 and b = 13. The first step is to substitute these values into the formula. This gives us C = √(24² + 13²). Next, we calculate the squares of a and b: 24² equals 576, and 13² equals 169. Now we substitute these squared values back into the equation: C = √(576 + 169). The next step is to add the two squared values together: 576 + 169 equals 745. So now our equation looks like this: C = √745. To find the value of C, we need to calculate the square root of 745. The square root of 745 is approximately 27.294688. Since we are asked to round the answer to two decimal places, we round 27.294688 to 27.29. Therefore, the length of the hypotenuse C is approximately 27.29 units.

Step-by-Step Calculation

Let's break down the calculation step-by-step for clarity. First, we have the formula for finding the hypotenuse: C = √(a² + b²). We are given a = 24 and b = 13. Substituting these values into the formula, we get C = √(24² + 13²). Next, we calculate the squares: 24² = 24 * 24 = 576, and 13² = 13 * 13 = 169. So, the equation becomes C = √(576 + 169). Now, we add the squares: 576 + 169 = 745. Thus, we have C = √745. Finally, we calculate the square root of 745, which is approximately 27.294688. Rounding this to two decimal places gives us 27.29. Therefore, the length of side C, the hypotenuse, is approximately 27.29 units. This step-by-step approach ensures accuracy and clarity in solving the problem, making it easier to understand the application of the Pythagorean theorem.

Rounding to Two Decimal Places

Rounding to two decimal places is a common practice in mathematics and various scientific and engineering fields to provide a level of precision that is both practical and meaningful. In our calculation of C, we arrived at an approximate value of 27.294688. To round this number to two decimal places, we focus on the first two digits after the decimal point, which are 29. The next digit, the third digit after the decimal point, is 4. The rule for rounding states that if the digit following the desired number of decimal places is 5 or greater, we round up the last digit. If it is less than 5, we leave the last digit as it is. In this case, since 4 is less than 5, we do not round up. Therefore, 27.294688 rounded to two decimal places is 27.29. This level of precision is often sufficient for many practical applications, providing a balance between accuracy and simplicity. Understanding rounding rules is crucial for ensuring the accuracy and reliability of calculations in various contexts.

Conclusion

In conclusion, by applying the Pythagorean theorem and following the formula C = √(a² + b²) with the given values of a = 24 and b = 13, we successfully calculated the length of the hypotenuse C. The step-by-step process involved substituting the given values into the formula, calculating the squares, adding them together, and then finding the square root. The result, approximately 27.294688, was then rounded to two decimal places, giving us a final answer of 27.29. This exercise demonstrates the practical application of the Pythagorean theorem in finding the unknown side of a right triangle. The ability to accurately apply this theorem is essential in various fields, including mathematics, engineering, and physics, highlighting the importance of understanding and mastering this fundamental concept. The process of rounding to two decimal places further refines the result to a level of precision suitable for many real-world applications.