Finding Missing Side Of A Right Triangle Pythagorean Theorem Explained

In the realm of mathematics, particularly in geometry, the Pythagorean Theorem stands as a cornerstone principle for understanding the relationships between the sides of a right triangle. This theorem, named after the ancient Greek mathematician Pythagoras, offers a straightforward yet powerful method for determining the length of an unknown side if the lengths of the other two sides are known. This article delves into the application of the Pythagorean Theorem to solve a specific problem: finding the length of a side in a right triangle where two sides are given as 5 cm and 13 cm. We'll explore the theorem itself, how to apply it in this scenario, and the step-by-step process to arrive at the solution. Understanding the Pythagorean Theorem is not only crucial for academic pursuits in mathematics but also for various real-world applications, from construction and engineering to navigation and even computer graphics. Its elegance and utility make it a fundamental concept in the field of mathematics, and mastering it opens the door to solving a wide range of geometric problems. Therefore, let's embark on this journey to understand and apply the Pythagorean Theorem to find the missing side of our right triangle, solidifying our understanding of this essential mathematical principle.

Understanding the Pythagorean Theorem

At its core, the Pythagorean Theorem describes a fundamental relationship within right triangles. A right triangle, by definition, is a triangle that contains one angle of 90 degrees – a right angle. The side opposite this right angle is called the hypotenuse, which is always the longest side of the triangle. The other two sides, which form the right angle, are referred to as the legs or cathetus of the triangle. The Pythagorean Theorem states that the square of the length of the hypotenuse (often denoted as 'c') is equal to the sum of the squares of the lengths of the other two sides (often denoted as 'a' and 'b'). This relationship is succinctly expressed by the equation: a² + b² = c². This seemingly simple equation is the key to unlocking a multitude of geometric problems involving right triangles. It allows us to calculate the length of any one side if we know the lengths of the other two, making it an invaluable tool in various fields. The theorem's significance extends beyond simple calculations; it forms the basis for many other geometric principles and constructions. Understanding the theorem's implications and applications is crucial for anyone studying mathematics, physics, engineering, or any field that relies on spatial reasoning. The beauty of the Pythagorean Theorem lies in its ability to connect the seemingly disparate concepts of lengths and squares, providing a powerful and elegant way to analyze the geometry of right triangles. This foundational understanding will pave the way for us to tackle the problem at hand: finding the missing side of a right triangle with given side lengths.

Applying the Theorem to the Problem

Now, let's apply the Pythagorean Theorem to the specific problem presented. We are given a right triangle with two sides measuring 5 cm and 13 cm. Our goal is to find the length of the third side. The first step is to identify which sides we know: the hypotenuse and one of the legs. In this case, 13 cm is the length of the hypotenuse (c), as it's the longest side, and 5 cm is the length of one of the legs (let's call it 'a'). We need to find the length of the other leg, which we'll call 'b'. Plugging the known values into the Pythagorean Theorem equation, a² + b² = c², we get: 5² + b² = 13². This equation now sets the stage for us to solve for 'b', the unknown side. To do this, we need to isolate 'b²' on one side of the equation. First, we calculate the squares of the known values: 5² = 25 and 13² = 169. Substituting these values back into the equation, we have: 25 + b² = 169. Now, we subtract 25 from both sides of the equation to isolate 'b²': b² = 169 - 25, which simplifies to b² = 144. This brings us to the final step of solving for 'b'. We need to find the square root of both sides of the equation to find the value of 'b'. The square root of 144 is 12. Therefore, b = 12 cm. This means the length of the missing side of the right triangle is 12 cm. By carefully applying the Pythagorean Theorem and following the algebraic steps, we have successfully determined the length of the unknown side.

Step-by-Step Solution

To further clarify the process, let's break down the solution into a clear, step-by-step guide. This will reinforce our understanding and provide a template for solving similar problems in the future. 1. Identify the Right Triangle and Known Sides: We have a right triangle, and we know two sides: one leg (a = 5 cm) and the hypotenuse (c = 13 cm). 2. State the Pythagorean Theorem: The Pythagorean Theorem is a² + b² = c². This is the foundation of our solution. 3. Substitute the Known Values: Plug the values of 'a' and 'c' into the equation: 5² + b² = 13². This gives us the specific equation we need to solve. 4. Calculate the Squares: Calculate the squares of the known values: 5² = 25 and 13² = 169. Our equation now looks like this: 25 + b² = 169. 5. Isolate the Unknown (b²): Subtract 25 from both sides of the equation to isolate b²: b² = 169 - 25. This simplifies to b² = 144. 6. Find the Square Root: Take the square root of both sides of the equation to solve for 'b': √b² = √144. This gives us b = 12. 7. State the Answer: The length of the missing side (b) is 12 cm. By following these steps, we can confidently apply the Pythagorean Theorem to solve for any missing side in a right triangle, given the lengths of the other two sides. This systematic approach not only leads to the correct answer but also helps solidify our understanding of the theorem and its applications. This step-by-step solution underscores the importance of a structured approach to problem-solving in mathematics.

Solution and Answer

Therefore, after meticulously applying the Pythagorean Theorem and following the step-by-step solution, we have arrived at the answer. The length of the missing side of the right triangle, which we denoted as 'b', is 12 cm. This corresponds to option B in the provided options. Our journey through the problem began with understanding the fundamental Pythagorean Theorem, which establishes the relationship between the sides of a right triangle. We then translated the problem into a mathematical equation, substituting the given values for the known sides. The subsequent steps involved algebraic manipulation to isolate the unknown variable and ultimately find its value. The solution highlights the power of the Pythagorean Theorem as a tool for solving geometric problems. It also demonstrates the importance of careful calculation and attention to detail in mathematics. The ability to correctly apply this theorem is a valuable skill in various fields, from construction and engineering to more abstract mathematical pursuits. By understanding the underlying principles and practicing problem-solving techniques, we can confidently tackle similar challenges in the future. The answer, 12 cm, not only completes this specific problem but also reinforces our understanding of the Pythagorean Theorem and its practical applications. This knowledge forms a solid foundation for further exploration in geometry and related mathematical disciplines.

Conclusion

In conclusion, we have successfully navigated the problem of finding the missing side of a right triangle using the Pythagorean Theorem. This exploration has not only provided us with a specific answer – 12 cm – but has also reinforced our understanding of a fundamental mathematical principle. The Pythagorean Theorem, with its elegant equation a² + b² = c², serves as a powerful tool for analyzing right triangles and solving for unknown side lengths. We have seen how to apply this theorem systematically, breaking down the problem into manageable steps: identifying the known sides, substituting values into the equation, isolating the unknown, and calculating the final result. This step-by-step approach is a valuable technique for problem-solving in mathematics and beyond. The significance of the Pythagorean Theorem extends far beyond the classroom. It is a cornerstone of geometry and has practical applications in various fields, including construction, navigation, and engineering. A firm grasp of this theorem is essential for anyone pursuing studies in these areas. Moreover, the problem-solving skills honed through applying the Pythagorean Theorem – critical thinking, logical reasoning, and attention to detail – are valuable assets in any endeavor. As we conclude this exploration, we recognize the importance of continuous learning and practice in mathematics. The Pythagorean Theorem is just one piece of a vast and interconnected field. By building a strong foundation in fundamental concepts, we can unlock more advanced topics and tackle increasingly complex challenges. Therefore, let us continue to explore the world of mathematics with curiosity and diligence, always striving to deepen our understanding and expand our problem-solving abilities.