Triangle Inequality Theorem Side Length Problem Explained

Chang knows one side of a triangle is 13 cm. The challenge lies in determining which set of two sides from the options provided could possibly form a triangle alongside the given side. This problem delves into a fundamental concept in geometry: the Triangle Inequality Theorem. This theorem is the key to solving this problem, so let's explore it in detail and apply it to Chang's challenge.

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This seemingly simple rule is the bedrock of triangle formation. If this condition isn't met, the sides simply cannot connect to form a closed figure, and thus, a triangle is impossible. To fully grasp this, imagine trying to construct a triangle with sticks of lengths 1, 2, and 5 units. No matter how you arrange them, the 1 and 2 unit sticks will never be able to bridge the gap created by the 5 unit stick. This is because 1 + 2 is less than 5, violating the theorem. On the other hand, if you had sticks of lengths 3, 4, and 5, you'd find that they perfectly form a triangle because 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3. The theorem holds true for all possible combinations of sides. This understanding is crucial for tackling problems like Chang's, where we need to assess the feasibility of different side length combinations. It's not just about the individual lengths but the relationship between them that dictates whether a triangle can exist. Think of it as a balancing act – the sides need to be in the right proportion to support the triangular structure. So, when faced with a question about triangle side lengths, always remember the Triangle Inequality Theorem as your guiding principle. It's the ultimate test for triangular possibility.

Applying the Triangle Inequality Theorem to the Problem

Now, let's apply the Triangle Inequality Theorem to Chang's problem. We know one side is 13 cm. We need to check each option to see if the sum of the two given sides is greater than 13 cm, and also ensure that the sum of 13 cm and each of the given sides is greater than the other given side. This is because the theorem must hold true for all three combinations of sides. In essence, we're not just looking for one condition to be satisfied; we need a harmonious relationship between all three sides. Imagine it as a three-way tug-of-war – each side needs to be strong enough to counter the combined force of the other two. If one side is too weak, the balance is disrupted, and the triangle collapses. This comprehensive check is vital to avoid overlooking any potential violations of the theorem. It’s not enough to simply see if two sides add up to more than 13; we need to ensure the theorem holds for all permutations. This rigorous approach ensures we identify the only set of sides that can definitively form a triangle with the given 13 cm side. This thoroughness is what distinguishes a correct answer from a hasty guess, and it's the key to mastering the application of the Triangle Inequality Theorem in various geometric problems. So, let's proceed with this meticulous check for each option, ensuring no potential violation goes unnoticed.

Evaluating Option A: 5 cm and 8 cm

Let's consider option A, which suggests the other two sides are 5 cm and 8 cm. We need to verify if these lengths, along with the given 13 cm, satisfy the Triangle Inequality Theorem. First, we check if 5 cm + 8 cm > 13 cm. This gives us 13 cm > 13 cm, which is false. The sum of these two sides is not greater than the third side. This single violation is enough to disqualify this set of sides. Remember, the Triangle Inequality Theorem is a strict condition – it must hold true for all combinations of sides. If even one combination fails, the entire set is invalid. It's like a chain; if one link is weak, the entire chain breaks. In this case, the 5 cm and 8 cm sides are simply too short to bridge the gap created by the 13 cm side. They lack the combined length necessary to form a closed triangular shape. Therefore, we can confidently eliminate option A without needing to check the other combinations. This illustrates the power of the theorem – a single violation can immediately rule out a possibility, saving us time and effort. It's a testament to the efficiency of mathematical principles in problem-solving.

Evaluating Option B: 6 cm and 7 cm

Now, let's examine option B, with side lengths of 6 cm and 7 cm. Again, we must rigorously apply the Triangle Inequality Theorem. We start by checking if 6 cm + 7 cm > 13 cm. This yields 13 cm > 13 cm, which, as before, is false. The sum of the proposed sides is not strictly greater than the given side. This immediately disqualifies option B. Just like in the previous case, the failure of one combination to satisfy the theorem is sufficient to rule out the entire set. The 6 cm and 7 cm sides, even combined, are not long enough to meet the requirement for forming a triangle with the 13 cm side. This consistent application of the theorem highlights its power as a definitive test for triangle viability. It's not a matter of approximation or close estimates; the condition must be unequivocally met. This example reinforces the importance of a thorough understanding of the theorem's requirements. We don't need to delve into further combinations; the initial violation is conclusive. Option B, therefore, cannot be the correct answer.

Evaluating Option C: 7 cm and 2 cm

Moving on to option C, the proposed side lengths are 7 cm and 2 cm. Let's put them to the Triangle Inequality Theorem test. The first check: 7 cm + 2 cm > 13 cm. This results in 9 cm > 13 cm, which is clearly false. The sum of 7 cm and 2 cm is less than 13 cm. This is a direct violation of the theorem, and thus, option C is immediately eliminated. The sides are simply too short to form a closed triangle with the 13 cm side. This quick determination demonstrates the efficiency of the theorem in identifying impossible triangle configurations. We don't need to explore other combinations; the initial failure is conclusive evidence against this option. It's a clear illustration of how mathematical principles can provide definitive answers with minimal calculation. The Triangle Inequality Theorem acts as a filter, quickly sifting out invalid possibilities and guiding us towards the correct solution.

Evaluating Option D: 8 cm and 9 cm

Finally, let's analyze option D, which suggests side lengths of 8 cm and 9 cm. We will meticulously apply the Triangle Inequality Theorem to see if this set is viable. First, we check if 8 cm + 9 cm > 13 cm. This gives us 17 cm > 13 cm, which is true. This is a promising start, but we can't stop here. We need to check all three combinations. Next, we check if 8 cm + 13 cm > 9 cm. This gives us 21 cm > 9 cm, which is also true. Lastly, we need to confirm if 9 cm + 13 cm > 8 cm. This yields 22 cm > 8 cm, which is, again, true. Since all three combinations satisfy the Triangle Inequality Theorem, option D is the correct answer. The sides 8 cm, 9 cm, and 13 cm can indeed form a triangle. This comprehensive check underscores the importance of verifying all conditions of the theorem. It's not enough to have one or two combinations work; all must hold true for the set to be valid. This rigorous approach ensures we arrive at a definitive and accurate answer. Option D, therefore, stands as the solution to Chang's challenge.

Conclusion: Option D (8 cm and 9 cm) is the Only Possible Set

In conclusion, after meticulously applying the Triangle Inequality Theorem to each option, we've determined that only option D, with side lengths of 8 cm and 9 cm, is possible alongside the given 13 cm side. Options A, B, and C failed to meet the theorem's requirement that the sum of any two sides must be greater than the third side. This exercise demonstrates the critical role of the Triangle Inequality Theorem in determining the feasibility of triangle formation. It's a fundamental principle in geometry that allows us to quickly and accurately assess whether a given set of side lengths can actually form a triangle. By understanding and applying this theorem, we can confidently solve problems like Chang's and deepen our understanding of geometric relationships. The power of mathematical theorems lies in their ability to provide definitive answers, and the Triangle Inequality Theorem is a prime example of this. It's a tool that empowers us to reason logically and solve problems with precision, making it an invaluable asset in the study of geometry.

Keywords

Triangle Inequality Theorem, side lengths, triangle, geometry, possible sides, Chang's problem