Identifying Right Triangle Side Lengths Using The Pythagorean Theorem

The core concept for determining if a set of numbers can represent the side lengths of a right triangle lies in the Pythagorean Theorem. This fundamental theorem of geometry states that in a right-angled 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 is expressed as a² + b² = c², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. The hypotenuse is always the longest side of the right triangle. This principle is the cornerstone for solving many geometric problems, especially those involving triangles and distances. Applying the Pythagorean Theorem effectively requires a clear understanding of its components and how they relate to each other within a right triangle. The theorem not only helps in identifying right triangles but also in calculating the lengths of unknown sides when the other sides are known. Therefore, mastering this theorem is crucial for anyone studying geometry or related fields. In practical applications, the Pythagorean Theorem is used extensively in fields such as construction, navigation, and engineering. Understanding this theorem also lays the groundwork for more advanced geometric concepts and problem-solving techniques. The elegance of the theorem lies in its simplicity and its wide range of applicability, making it an essential tool in the mathematical arsenal. Therefore, when faced with questions about right triangles, the Pythagorean Theorem should be the first principle to consider. Let's delve deeper into how we can use this theorem to solve the problem at hand and understand why certain sets of numbers qualify while others do not. By thoroughly grasping the implications and applications of the Pythagorean Theorem, you can confidently tackle various geometric challenges.

To determine which set of numbers can represent the side lengths of a right triangle, we need to apply the Pythagorean Theorem to each option. This involves identifying the longest side in each set, which would be the potential hypotenuse, and then checking if the square of the longest side is equal to the sum of the squares of the other two sides. This process is crucial in verifying the right triangle condition. We will go through each option systematically, performing the necessary calculations and comparisons. This methodical approach ensures accuracy and helps in understanding why some sets satisfy the theorem while others do not. The key is to correctly identify the potential hypotenuse and apply the formula a² + b² = c² accurately. Let's break down each option: A, B, C, and D, and see which one holds true according to the Pythagorean Theorem. This detailed analysis will not only provide the correct answer but also reinforce the understanding of the theorem itself. It is important to note that the order of 'a' and 'b' does not matter, as addition is commutative, but 'c' must always be the longest side. This careful application of the theorem is fundamental to solving problems related to right triangles and is a key skill in geometry. Furthermore, this process demonstrates the practical application of the theorem and its importance in identifying and understanding right-angled triangles. Remember, only the set of numbers that satisfies the equation a² + b² = c² can represent the sides of a right triangle.

• Option A: 8, 12, 15

  In this set, 15 is the longest side. We need to check if 8² + 12² = 15². Calculating the squares, we get 64 + 144 = 225. Adding 64 and 144 gives us 208. Since 208 is not equal to 225, this set of numbers does not represent the sides of a right triangle. The discrepancy between the sum of the squares of the two shorter sides and the square of the longest side indicates that the triangle formed by these lengths would not have a right angle. This is a clear example of how the Pythagorean Theorem can be used to verify the properties of triangles. Therefore, we can confidently eliminate this option as a potential answer. The calculation clearly demonstrates that the numbers do not fit the required relationship for a right-angled triangle. This methodical approach is crucial in solving such problems and understanding the underlying principles of geometry. Furthermore, it emphasizes the importance of accurate calculation in mathematical problem-solving. By carefully examining each step, we can ensure that our conclusions are well-supported by the evidence.

• Option B: 10, 24, 26

  Here, 26 is the longest side. We need to verify if 10² + 24² = 26². Calculating the squares, we have 100 + 576 = 676. Adding 100 and 576 indeed gives us 676. Since 676 is equal to 676, this set of numbers does represent the sides of a right triangle. This confirms that a triangle with sides 10, 24, and 26 would have a right angle, satisfying the Pythagorean Theorem. This positive result highlights the theorem's power in identifying right triangles. The precise equality between the sum of the squares of the two shorter sides and the square of the longest side provides definitive proof. This option serves as a clear example of how the Pythagorean Theorem works in practice. It is important to note that this set of numbers forms a Pythagorean triple, which is a set of three positive integers that satisfy the equation a² + b² = c². Therefore, we have successfully identified a set of numbers that fulfills the criteria. This thorough verification process ensures the accuracy of our conclusion and reinforces the understanding of the theorem's application. The clarity of this result emphasizes the importance of the Pythagorean Theorem in geometric problem-solving.

• Option C: 12, 20, 25

  In this option, 25 is the longest side. We need to check if 12² + 20² = 25². Calculating the squares, we get 144 + 400 = 625. Adding 144 and 400 gives us 544. Since 544 is not equal to 625, this set of numbers does not represent the sides of a right triangle. This discrepancy indicates that the triangle formed by these lengths would not have a right angle. The failure to satisfy the Pythagorean Theorem clearly disqualifies this option. This further reinforces the theorem's role in determining the properties of triangles. The calculations demonstrate that the relationship between the sides does not meet the requirements for a right-angled triangle. Therefore, we can confidently eliminate this option as a potential answer. The methodical application of the theorem allows us to quickly identify non-right triangles. This process underscores the importance of accurate arithmetic in mathematical problem-solving. Each step in the calculation is crucial for reaching a valid conclusion. By carefully examining the results, we can ensure that our answers are well-founded.

• Option D: 15, 18, 20

  Here, 20 is the longest side. We need to verify if 15² + 18² = 20². Calculating the squares, we have 225 + 324 = 400. Adding 225 and 324 gives us 549. Since 549 is not equal to 400, this set of numbers does not represent the sides of a right triangle. This indicates that a triangle with these side lengths would not have a right angle, as the Pythagorean Theorem is not satisfied. This negative result further illustrates the theorem's effectiveness in identifying right triangles. The significant difference between the sum of the squares of the two shorter sides and the square of the longest side clearly demonstrates the lack of a right angle. Therefore, we can confidently eliminate this option from consideration. The systematic application of the Pythagorean Theorem allows us to quickly determine the validity of triangle side lengths. This process highlights the importance of mathematical rigor in problem-solving. Each calculation contributes to the overall conclusion, ensuring that our answer is accurate and well-supported. By carefully analyzing the results, we gain a deeper understanding of the theorem's implications and its applications in geometry.

After analyzing all the options using the Pythagorean Theorem, we found that only one set of numbers, 10, 24, and 26 (Option B), satisfies the condition a² + b² = c². This confirms that Option B is the correct answer, as these numbers can indeed represent the side lengths of a right triangle. The other options failed to satisfy the theorem, indicating that they cannot form right triangles. This exercise highlights the importance and applicability of the Pythagorean Theorem in determining the properties of triangles. The systematic approach of applying the theorem to each option ensures accuracy and a clear understanding of the solution. This problem-solving process demonstrates the practical use of mathematical principles in geometry. Therefore, the ability to apply the Pythagorean Theorem is crucial for anyone studying mathematics or related fields. The theorem not only helps in identifying right triangles but also in solving various geometric problems. The clarity of the result underscores the fundamental nature of the Pythagorean Theorem in geometry and its wide range of applications. By mastering this theorem, one can confidently tackle various challenges related to triangles and their properties. The consistent application of the theorem leads to a definitive answer, reinforcing the importance of mathematical rigor in problem-solving. Therefore, the correct answer is Option B, as it is the only set of numbers that can represent the side lengths of a right triangle.

Final Answer: B. 10, 24, 26