Pythagorean Theorem Problem Length Of A Right Triangle Leg
The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. This theorem is extremely useful and has wide applications in various fields, including architecture, engineering, and navigation. The theorem states that in a right triangle, 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 (legs). Mathematically, this relationship is expressed as: a² + b² = c², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.
In this article, we will explore a specific problem that involves using the Pythagorean theorem to find the length of a leg of a right triangle when the length of the other leg and the hypotenuse are known. This type of problem often appears in math textbooks and standardized tests, and mastering its solution is crucial for a solid understanding of geometry. Let's delve into the problem-solving process step by step, ensuring that each concept is clear and easily understood.
Problem Statement
The problem we are tackling is as follows: Using the Pythagorean theorem, find the length of a leg of a right triangle if the other leg is 8 feet long and the hypotenuse is 10 feet long. This problem provides us with the lengths of one leg and the hypotenuse and asks us to calculate the length of the remaining leg. The key to solving this problem lies in correctly applying the Pythagorean theorem and algebraic manipulation.
Before we begin the solution, let's visualize the right triangle. Imagine a triangle with one angle measuring 90 degrees. The longest side, the hypotenuse, stretches across from the right angle and measures 10 feet. One of the shorter sides, a leg, is 8 feet long. Our task is to find the length of the other leg. This visualization helps in setting up the equation and understanding the relationship between the sides.
Now, let's proceed with the step-by-step solution, breaking down each stage to ensure clarity and accuracy.
Step-by-Step Solution
Step 1: Understand the Pythagorean Theorem
The Pythagorean theorem, as mentioned earlier, is expressed as a² + b² = c², where 'a' and 'b' are the lengths of the legs of the right triangle, and 'c' is the length of the hypotenuse. In our problem, we are given the length of one leg (let's call it 'a') as 8 feet and the length of the hypotenuse ('c') as 10 feet. We need to find the length of the other leg ('b').
Understanding this theorem is the cornerstone of solving this problem. It's essential to recognize that this theorem applies exclusively to right triangles and provides a direct relationship between the lengths of the sides. The hypotenuse is always the longest side and is opposite the right angle. The legs are the two shorter sides that form the right angle.
Step 2: Substitute the Given Values into the Formula
We substitute the given values into the Pythagorean theorem formula: a² + b² = c². In our case, a = 8 feet and c = 10 feet. Plugging these values into the formula, we get:
8² + b² = 10²
This substitution is a crucial step because it translates the geometric problem into an algebraic equation that we can solve. It is important to ensure that the values are placed correctly in the formula. The leg 'a' is squared, the unknown leg 'b' is squared, and the hypotenuse 'c' is squared. This setup allows us to isolate the unknown variable and find its value.
Step 3: Simplify the Equation
Next, we simplify the equation by calculating the squares of the known values:
8² = 64
10² = 100
Substituting these values back into the equation, we get:
64 + b² = 100
Simplifying the squares is a straightforward arithmetic operation but is critical for progressing towards the solution. Ensuring that these calculations are accurate prevents errors in the subsequent steps. The equation now represents a simpler form that allows us to isolate the unknown variable, 'b²'.
Step 4: Isolate the Unknown Variable
To isolate b², we subtract 64 from both sides of the equation:
64 + b² - 64 = 100 - 64
This simplifies to:
b² = 36
The goal of this step is to get b² by itself on one side of the equation. Subtracting 64 from both sides maintains the equation's balance and allows us to isolate the term containing the unknown variable. The result, b² = 36, brings us closer to finding the value of 'b'.
Step 5: Solve for the Unknown Leg
To find the value of 'b', we take the square root of both sides of the equation:
√b² = √36
This gives us:
b = 6
Since we are dealing with lengths, we only consider the positive square root. Therefore, the length of the unknown leg is 6 feet.
Taking the square root is the final step in solving for 'b'. It undoes the squaring operation and reveals the length of the unknown leg. Remembering to consider only the positive root in the context of lengths is important. The solution, b = 6 feet, represents the answer to the problem.
Final Answer
Therefore, the length of the other leg of the right triangle is 6 feet. This corresponds to answer choice C. 6 ft.
This final answer is the culmination of the step-by-step process we have followed. It answers the question posed in the problem statement and provides a definitive solution. Verifying the answer by plugging it back into the original equation can further ensure its correctness.
Conclusion
In conclusion, using the Pythagorean theorem, we successfully found the length of a leg of a right triangle when the other leg and the hypotenuse were given. The key steps involved understanding the theorem, substituting the given values into the formula, simplifying the equation, isolating the unknown variable, and solving for the unknown leg. This problem-solving process highlights the importance of a strong foundation in algebraic manipulation and geometric principles.
The Pythagorean theorem is a cornerstone of geometry and has numerous applications in real-world scenarios. Mastering the application of this theorem not only helps in solving mathematical problems but also enhances analytical and problem-solving skills that are valuable in various fields. Understanding how to manipulate equations and apply mathematical principles to geometric problems is crucial for further studies in mathematics and related disciplines.
This problem serves as an excellent example of how abstract mathematical concepts can be applied to solve concrete problems. By breaking down the problem into manageable steps, we were able to clearly demonstrate the solution process. This approach can be applied to other mathematical problems as well, fostering a systematic and logical method of problem-solving. Remember, the key to success in mathematics is understanding the underlying principles and practicing their application through various examples.
By working through this problem, we have not only found the length of the unknown leg but also reinforced our understanding of the Pythagorean theorem and its applications. This knowledge will undoubtedly be beneficial in tackling more complex geometric problems in the future.
In summary, the process of solving this problem has demonstrated the power and utility of the Pythagorean theorem in determining the relationships between the sides of a right triangle. It underscores the importance of understanding fundamental mathematical principles and applying them systematically to solve problems effectively. The ability to approach mathematical challenges with a clear, step-by-step methodology is a skill that is highly valuable in both academic and practical settings.
