Determining Right Triangles Using The Pythagorean Theorem

Jul 11, 2025 by ADMIN 58 views

Determine Which Set of Side Measurements Could Be Used to Form a Right Triangle

Determining whether a set of side measurements can form a right triangle is a fundamental concept in geometry, rooted in the Pythagorean Theorem. This theorem, a cornerstone of Euclidean geometry, provides a straightforward method to verify the relationship between the sides of a right triangle. In this comprehensive guide, we will delve into the intricacies of the Pythagorean Theorem, explore its applications, and meticulously analyze the given sets of side measurements to ascertain which one(s) satisfy the conditions for forming a right triangle. Our focus will be on providing a clear, step-by-step approach, ensuring a thorough understanding of the underlying principles and calculations involved. By the end of this exploration, you will be equipped with the knowledge and skills to confidently identify right triangles based on their side lengths, a crucial ability in various mathematical and real-world contexts.

Understanding the Pythagorean Theorem

At the heart of determining right triangles lies the Pythagorean Theorem. This theorem 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 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. This seemingly simple equation is a powerful tool for verifying the validity of right triangles and solving for unknown side lengths. The theorem is not just a mathematical abstraction; it has practical applications in various fields, including construction, navigation, and engineering. For instance, architects use the Pythagorean Theorem to ensure that building corners are perfectly square, while surveyors rely on it for accurate land measurements. Understanding the theorem's implications and applications is crucial for mastering geometry and its related fields. In the following sections, we will apply this theorem to the given sets of side measurements, rigorously testing each set to determine if it satisfies the fundamental condition for forming a right triangle.

Applying the Theorem to the Given Sets

To determine which set of side measurements can form a right triangle, we must apply the Pythagorean Theorem to each set individually. This involves identifying the potential hypotenuse (the longest side) and then verifying if the sum of the squares of the shorter sides equals the square of the longest side. This process requires careful calculation and attention to detail to avoid errors. We will start with the first set, meticulously calculate the squares of each side, and then compare the sums. If the equation a² + b² = c² holds true, then the set forms a right triangle; otherwise, it does not. We will repeat this procedure for each of the remaining sets, providing a clear and concise explanation for each step. This systematic approach ensures that we thoroughly examine each set of measurements and arrive at an accurate conclusion. By demonstrating the application of the Pythagorean Theorem in a step-by-step manner, we aim to solidify your understanding of this critical concept and enhance your ability to solve similar problems independently. The following subsections will detail the calculations for each set, highlighting the key steps and results.

Analyzing the Sets of Side Measurements

Now, let's apply the Pythagorean Theorem to each set of side measurements provided. We will meticulously analyze each set to determine if it satisfies the condition a² + b² = c², where 'c' represents the potential hypotenuse (the longest side). This process involves squaring each side length, summing the squares of the two shorter sides, and comparing the result to the square of the longest side. A detailed explanation for each set will be provided to illustrate the application of the theorem and the reasoning behind our conclusions. This step-by-step approach is crucial for understanding how the Pythagorean Theorem works in practice and for developing problem-solving skills in geometry. We will begin with the first set and proceed sequentially, ensuring that each set is thoroughly examined before moving on to the next. Our goal is not only to identify the sets that form right triangles but also to explain why certain sets do not meet the criteria.

Set 1: √19, √35, 54

For the first set, the side measurements are √19, √35, and 54. To determine if these measurements could form a right triangle, we need to identify the longest side, which is 54, and then verify if (√19)² + (√35)² = 54². Let's calculate each part of the equation:

(√19)² = 19
(√35)² = 35
54² = 2916

Now, we add the squares of the two shorter sides: 19 + 35 = 54. Comparing this sum to the square of the longest side, we see that 54 ≠ 2916. Therefore, this set of side measurements cannot form a right triangle because it does not satisfy the Pythagorean Theorem. The vast discrepancy between the sum of the squares of the shorter sides and the square of the longest side clearly indicates that this triangle cannot have a right angle. This example demonstrates the importance of precise calculations and comparisons when applying the Pythagorean Theorem.

Set 2: √15, 6, √51

Moving on to the second set, we have side measurements of √15, 6, and √51. The potential hypotenuse is √51 since it's the longest side. We need to check if (√15)² + 6² = (√51)². Let's calculate each term:

(√15)² = 15
6² = 36
(√51)² = 51

Now, we add the squares of the two shorter sides: 15 + 36 = 51. Comparing this to the square of the longest side, we find that 51 = 51. This equality confirms that the set of side measurements can form a right triangle because it perfectly satisfies the Pythagorean Theorem. This example illustrates how the Pythagorean Theorem can be used to verify the right-angled nature of a triangle when given its side lengths. The precise match between the sum of the squares of the shorter sides and the square of the longest side leaves no doubt about the validity of this right triangle.

Set 3: 5, 8, 30

For the third set of side measurements, we have 5, 8, and 30. The longest side is 30, so we need to verify if 5² + 8² = 30². Calculating each term:

5² = 25
8² = 64
30² = 900

Now, we add the squares of the two shorter sides: 25 + 64 = 89. Comparing this to the square of the longest side, we see that 89 ≠ 900. This significant difference indicates that the set of side measurements cannot form a right triangle. The sum of the squares of the two shorter sides is far less than the square of the longest side, invalidating the Pythagorean relationship. This example further reinforces the importance of the Pythagorean Theorem as a tool for identifying right triangles and demonstrates how substantial discrepancies can quickly rule out a set of measurements.

Set 4: 5, 6, 7

Finally, for the fourth set, we have side measurements of 5, 6, and 7. The longest side is 7, so we need to check if 5² + 6² = 7². Calculating each term:

5² = 25
6² = 36
7² = 49

Adding the squares of the two shorter sides, we get 25 + 36 = 61. Comparing this to the square of the longest side, we see that 61 ≠ 49. This inequality means that the set of side measurements cannot form a right triangle. The sum of the squares of the two shorter sides is greater than the square of the longest side, which violates the Pythagorean Theorem. This final example illustrates that even seemingly close values must precisely satisfy the theorem's equation to qualify as a right triangle.

Conclusion

In conclusion, after meticulously analyzing each set of side measurements using the Pythagorean Theorem, we have determined that only the set √15, 6, √51 can form a right triangle. This set satisfied the condition a² + b² = c², demonstrating the precise relationship between the sides of a right-angled triangle. The other sets failed to meet this criterion, highlighting the importance of the Pythagorean Theorem as a definitive test for right triangles. This exercise underscores the fundamental role of the Pythagorean Theorem in geometry and its practical applications in various fields. Understanding and applying this theorem correctly is essential for solving problems related to right triangles and for developing a strong foundation in mathematical reasoning. By working through these examples, we have gained valuable insights into how to identify right triangles and appreciate the elegance and power of the Pythagorean Theorem.