Determining The Possible Length Of The Third Side Of A Triangle

Understanding the Triangle Inequality Theorem

When dealing with triangles, a fundamental principle governs the possible lengths of their sides: the Triangle Inequality Theorem. This 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 principle ensures that the triangle can actually be constructed; otherwise, the sides would simply not be able to connect to form a closed figure. In essence, this theorem provides a crucial constraint on the dimensions of triangles, ensuring their physical realizability. Think of it as a structural rule that dictates the very possibility of a triangle's existence.

To truly grasp the essence of this theorem, let’s delve deeper into its implications. Imagine you have two sticks, one 5 inches long and the other 12 inches long. If you try to form a triangle with a third stick that is, say, only 2 inches long, you'll find that the two shorter sticks won't be able to meet and close the triangle. The 2-inch stick is simply too short to bridge the gap between the ends of the 5-inch and 12-inch sticks. This is a direct consequence of the Triangle Inequality Theorem. Conversely, if the third stick is excessively long, such as 20 inches, it becomes clear that the 5-inch and 12-inch sticks, when placed end-to-end, will not be able to reach far enough to form a vertex with the 20-inch stick. The theorem, therefore, establishes both a lower and an upper bound on the possible lengths of the third side, ensuring that it is neither too short nor too long to complete the triangle. Understanding this balance is key to solving problems related to triangle side lengths.

Furthermore, the Triangle Inequality Theorem is not just a theoretical concept; it has practical applications in various fields, including engineering, architecture, and even computer graphics. Engineers use it to ensure the stability of structures, architects rely on it to design buildings with appropriate dimensions, and computer graphics designers employ it to create realistic 3D models. The theorem's widespread utility highlights its importance as a foundational principle in geometry and its relevance to real-world problem-solving. By mastering this theorem, you not only gain a deeper understanding of triangles but also equip yourself with a valuable tool for tackling a wide array of practical challenges. Its elegance lies in its simplicity and its power in its universality.

Applying the Theorem to the Problem

In this specific problem, we are given two sides of a triangle measuring 5 inches and 12 inches. Our task is to determine which of the provided options could represent the length of the third side. To do this effectively, we will apply the Triangle Inequality Theorem. This involves setting up inequalities to define the possible range of lengths for the third side. The theorem dictates that the third side must be shorter than the sum of the other two sides and longer than the difference between them. By carefully considering these two conditions, we can narrow down the possibilities and identify the correct answer from the given choices.

To begin, let's denote the length of the third side as 'x'. According to the Triangle Inequality Theorem, the following two inequalities must hold true:

1. The sum of the two given sides must be greater than the third side: 5 + 12 > x, which simplifies to 17 > x. This inequality tells us that the third side must be less than 17 inches. If the third side were 17 inches or longer, the two shorter sides (5 and 12 inches) would not be able to "reach" each other to form a closed triangle.
2. The difference between the two given sides must be less than the third side: |12 - 5| < x, which simplifies to 7 < x. This inequality tells us that the third side must be greater than 7 inches. If the third side were 7 inches or shorter, it would be too short to "bridge the gap" between the ends of the 5-inch and 12-inch sides.

Combining these two inequalities, we find that the length of the third side, x, must fall within the range 7 < x < 17. This means the third side must be greater than 7 inches and less than 17 inches. Now, we can use this range to evaluate the given options and determine which one satisfies the condition. By systematically checking each option against this range, we can confidently identify the correct answer, demonstrating a clear understanding of the Triangle Inequality Theorem and its practical application.

Evaluating the Answer Choices

Now that we've established the range for the possible length of the third side (7 < x < 17), let's evaluate the given answer choices:

• A. 3 in. 3 inches is less than 7 inches, so this option does not satisfy the inequality 7 < x. Therefore, a triangle with sides 5 inches, 12 inches, and 3 inches cannot exist.
• B. 6 in. 6 inches is also less than 7 inches, so this option similarly fails to meet the requirement 7 < x. A triangle with sides 5 inches, 12 inches, and 6 inches is not possible.
• C. 10 in. 10 inches falls within the range 7 < x < 17, as it is greater than 7 inches and less than 17 inches. Thus, this option could be the length of the third side, making it a potential answer.
• D. 18 in. 18 inches is greater than 17 inches, violating the inequality x < 17. A triangle with sides 5 inches, 12 inches, and 18 inches cannot be formed.

Based on this analysis, only option C, 10 inches, falls within the acceptable range for the third side. The other options either fail to meet the minimum length requirement (7 inches) or exceed the maximum length constraint (17 inches). This systematic evaluation clearly demonstrates the application of the Triangle Inequality Theorem in determining the feasibility of triangle side lengths. By methodically comparing each option against the established range, we can confidently arrive at the correct answer.

Conclusion: The Correct Answer

After carefully applying the Triangle Inequality Theorem and evaluating the answer choices, we can confidently conclude that the length of the third side could be 10 inches (Option C). This is the only option that satisfies both conditions of the theorem: it is greater than the difference between the other two sides (7 inches) and less than the sum of the other two sides (17 inches). Options A, B, and D fail to meet these criteria and, therefore, cannot represent the length of the third side of the triangle.

The Triangle Inequality Theorem is a fundamental principle in geometry that governs the possible lengths of a triangle's sides. It ensures that the sum of any two sides must be greater than the third side, preventing scenarios where the sides cannot connect to form a closed figure. This theorem is not only a theoretical concept but also has practical applications in various fields, including engineering and architecture. Understanding and applying this theorem is crucial for solving problems related to triangle side lengths and for gaining a deeper appreciation of geometric principles.

In this particular problem, by systematically applying the Triangle Inequality Theorem, we were able to narrow down the possibilities and identify the correct answer. This process highlights the importance of mastering fundamental concepts in mathematics, as they provide the foundation for solving more complex problems. The ability to apply theorems and principles effectively is a key skill in mathematics and is essential for success in related fields. Therefore, a thorough understanding of concepts like the Triangle Inequality Theorem is invaluable for anyone studying or working with geometry.