Calculate Hypotenuse 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 establishes a relationship between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). The formula representing this theorem is:

$C = \sqrt{a^2 + b^2}$

Where:

• C represents the length of the hypotenuse.
• a and b represent the lengths of the two legs of the right triangle.

We will apply this formula to a specific scenario where we are given the lengths of the two legs, a and b, and we need to calculate the length of the hypotenuse, C. We will also discuss the practical applications of the Pythagorean Theorem and its significance in various fields.

Applying the Formula

In our given scenario, we have the following information:

• a = 24
• b = 13

Our goal is to calculate the value of C using the Pythagorean Theorem formula. To do this, we will substitute the given values of a and b into the formula and perform the necessary calculations.

Step-by-Step Calculation

1. Substitute the values: Replace a with 24 and b with 13 in the formula: $C = \sqrt{24^2 + 13^2}$

2. Calculate the squares: Calculate the squares of 24 and 13: $24^2 = 576$ $13^2 = 169$

3. Add the squares: Add the squares obtained in the previous step: $576 + 169 = 745$

4. Take the square root: Calculate the square root of the sum: $C = \sqrt{745}$

5. Approximate the result: The square root of 745 is approximately 27.29.

Therefore, the length of the hypotenuse, C, is approximately 27.29 units.

Importance of Rounding

In many practical applications, it is essential to round the final answer to a reasonable number of decimal places. This is because measurements and calculations often involve approximations, and rounding helps to avoid unnecessary precision. In our case, we rounded the result to two decimal places, which is a common practice for engineering and scientific calculations.

Rounding rules dictate that if the digit following the desired decimal place 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 our example, the result of the square root was approximately 27.29487, so we rounded it to 27.29.

The Pythagorean Theorem in Real-World Applications

The Pythagorean Theorem isn't just a math equation; it's a powerful tool that has real-world applications across numerous fields. Understanding and applying this theorem can help solve practical problems in various scenarios. For example, let's consider construction. Builders use the theorem to ensure structures have square corners. Imagine constructing the foundation of a house. If the corners aren't perfectly square, the entire structure could be compromised. By using the theorem, builders can verify that the dimensions of the foundation form a right triangle, ensuring a solid, stable base. This precise application of mathematical principles is crucial in construction and engineering.

Navigation is another area where the theorem plays a vital role. Pilots and sailors use it to calculate the shortest distance, or displacement, between two points. This is particularly useful when dealing with routes that aren't direct. For instance, if a plane flies a certain distance east and then a certain distance north, the Pythagorean Theorem can determine the direct distance from the starting point. This helps in planning fuel consumption and estimating travel times, making navigation safer and more efficient. In essence, the theorem provides a mathematical shortcut to understanding spatial relationships.

Engineering also heavily relies on the Pythagorean Theorem. Structural engineers use it to design bridges and buildings, ensuring they can withstand various stresses and loads. When designing a bridge, engineers need to calculate the lengths and angles of support beams to guarantee stability. The theorem helps them determine how different structural components interact and distribute weight. This is critical in preventing structural failures and ensuring public safety. The theorem's application in engineering underscores the importance of mathematics in creating a safe and functional infrastructure.

Conclusion

In this article, we successfully calculated the length of the hypotenuse C using the Pythagorean Theorem formula: $C = \sqrt{a^2 + b^2}$. Given the values a = 24 and b = 13, we substituted these values into the formula, performed the calculations, and found that C is approximately 27.29 units.

This exercise demonstrates the practical application of the Pythagorean Theorem in solving real-world problems. The theorem is a fundamental concept in geometry and has numerous applications in fields such as construction, navigation, and engineering. Understanding and applying the Pythagorean Theorem is essential for anyone working in these fields, as it allows for accurate calculations and problem-solving.

We also discussed the importance of rounding in practical applications. Rounding helps to avoid unnecessary precision and ensures that the final answer is meaningful in the context of the problem. In our case, we rounded the result to two decimal places, which is a common practice for engineering and scientific calculations.

By mastering the Pythagorean Theorem and its applications, you will be well-equipped to tackle a wide range of problems in mathematics and beyond. This theorem serves as a cornerstone of geometric understanding and a valuable tool for problem-solving in various disciplines.