Triangle Inequality Theorem Determine Possible Side Lengths

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Hey guys! Let's dive into an interesting problem in geometry that involves figuring out the possible lengths for the sides of a triangle. Specifically, we're going to use the Triangle Inequality Theorem to determine the range of possible lengths for the third side of a triangle when we already know the lengths of the other two sides. It’s a fundamental concept that helps us understand how triangles are constructed and what their limitations are. So, buckle up, and let’s get started!

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem is a crucial concept in geometry that dictates the relationship between the sides of any triangle. In simple terms, it 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 rule holds true for all three combinations of sides within a triangle. Think of it this way: if you have two short sticks, they won't be able to meet to form a triangle with a very long stick. They simply won't reach! Mathematically, if we have a triangle with sides a, b, and c, the theorem can be expressed as three inequalities:

  1. a + b > c
  2. a + c > b
  3. b + c > a

These inequalities ensure that the sides can actually connect to form a closed figure, which is what a triangle is. Without this condition, the sides might be too short to meet, or one side might be too long, preventing the other two from connecting. To truly grasp this, imagine trying to build a triangle with sides of lengths 1, 2, and 5. You'll quickly realize that the sides with lengths 1 and 2 are too short to meet and form a triangle with the side of length 5. Now, let's apply this theorem to a specific problem to see how it works in practice. Understanding this theorem is key to solving a variety of geometric problems, from basic triangle constructions to more complex proofs and calculations. So, let’s keep this principle in mind as we move forward and tackle the challenge at hand. This theorem not only helps us in theoretical mathematics but also has practical applications in fields like engineering and architecture, where understanding structural stability is crucial. For example, when designing bridges or buildings, engineers must ensure that the triangular supports meet the criteria set by the Triangle Inequality Theorem to ensure the structure's integrity and stability. The beauty of this theorem lies in its simplicity and its profound implications. It’s a fundamental rule that governs the very existence of triangles, making it an indispensable tool in the world of geometry and beyond.

Applying the Theorem to Our Problem: Sides of Length 6 and 12

Alright, let’s get to the heart of the matter! We have a triangle with two sides that measure 6 and 12 units, respectively. Our mission, should we choose to accept it (and we do!), is to determine the possible range of lengths for the third side. Let's call the length of the third side "c." We're going to use the Triangle Inequality Theorem to figure this out. Remember, this theorem tells us that the sum of the lengths of any two sides must be greater than the length of the third side. So, we need to apply this rule to all possible combinations of the sides we have. First, let’s consider the combination of sides 6 and 12. According to the theorem, their sum must be greater than the third side, c. This gives us our first inequality: 6 + 12 > c, which simplifies to 18 > c. This tells us that the third side must be shorter than 18 units. Now, let’s look at another combination. We need to ensure that the sum of side 6 and the unknown side c is greater than the side with length 12. This gives us the inequality: 6 + c > 12. If we solve this inequality for c, we subtract 6 from both sides, resulting in c > 6. This means that the third side must be longer than 6 units. Lastly, we need to consider the combination of side 12 and the unknown side c. The sum of these two sides must be greater than the side with length 6. This gives us the inequality: 12 + c > 6. Solving for c, we subtract 12 from both sides, resulting in c > -6. However, since side lengths cannot be negative, this inequality doesn't provide a meaningful lower bound for the length of side c. So, we can disregard it in our final answer. By applying the Triangle Inequality Theorem, we’ve established two critical boundaries for the length of the third side. It must be less than 18 units (from the first inequality) and greater than 6 units (from the second inequality). This narrows down the possible lengths of the third side to a specific range, ensuring that the triangle can actually exist. Let's move on to putting these findings together to define the exact range.

Determining the Range of Possible Lengths

Okay, guys, we've done the groundwork! We know that the third side, which we've called "c," has to meet two important conditions based on the Triangle Inequality Theorem. First, we figured out that c must be less than 18 (c < 18). This is because the sum of the other two sides (6 and 12) is 18, and the third side has to be shorter than that for a triangle to actually form. Imagine trying to stretch a side to be as long as or longer than the combined length of the other two – it just wouldn't connect! Second, we found out that c has to be greater than 6 (c > 6). This is because the sum of the third side and the side with length 6 must be greater than the side with length 12. If c were 6 or less, it wouldn't be long enough to help the side of length 6 reach and connect with the side of length 12. So, we have our upper and lower limits. Now, how do we put these together to define the range of possible lengths for c? Well, we simply combine these two inequalities. We know that c must be greater than 6 AND less than 18. This can be written as a compound inequality: 6 < c < 18. This compound inequality tells us that the length of the third side must fall strictly between 6 and 18 units. It cannot be exactly 6 units, and it cannot be exactly 18 units – it has to be somewhere in between. To visualize this, you can think of a number line. Mark the points 6 and 18. The possible lengths for c are all the points on the line between 6 and 18, not including 6 and 18 themselves. So, any length within this range will allow us to form a valid triangle with the sides of length 6 and 12. This range gives us a clear understanding of the limitations and possibilities when constructing triangles, and it’s a direct application of the Triangle Inequality Theorem. Let’s take a look at some options and see which ones fit within this range.

Evaluating the Given Options

Now, let's put our newfound knowledge to the test and evaluate the options provided in the original problem. We’ve determined that the possible lengths for the third side (c) must fall within the range 6 < c < 18. This means that c has to be greater than 6 and less than 18. We'll go through each option and see if it aligns with this range.

  • Option A: 12 < c < 6

    This option states that the length of the third side (c) must be greater than 12 and less than 6. This is immediately problematic because it's impossible for a number to be simultaneously greater than 12 and less than 6. These conditions are contradictory, so this option is incorrect. Think of it like trying to be taller than 6 feet and shorter than 5 feet at the same time – it just can't happen!

  • Option B: -6 < c < -18

    This option suggests that the length of the third side falls between -6 and -18. We run into a major issue here because side lengths cannot be negative values. In geometry, lengths are always positive measurements. So, any option that includes negative values is not viable. Furthermore, just like in Option A, the conditions are contradictory. It’s impossible for a number to be greater than -18 and less than -6 at the same time. Therefore, this option is also incorrect.

  • Option C: 6 < c < 18

    This option states that the length of the third side (c) must be greater than 6 and less than 18. Hey, that sounds familiar! This is exactly the range we determined using the Triangle Inequality Theorem. We figured out that the third side has to be longer than 6 units and shorter than 18 units for the triangle to be valid. This option perfectly matches our calculated range, making it the correct answer. So, after carefully analyzing the options and applying the Triangle Inequality Theorem, we've successfully identified the correct range for the third side. Let’s wrap things up and summarize our findings.

Conclusion: The Correct Range for the Third Side

Alright, guys, we've reached the finish line! Let's recap what we've accomplished in this geometrical journey. We started with a triangle that had two known sides, measuring 6 and 12 units. Our mission was to figure out the possible range of lengths for the third side. To do this, we dove headfirst into the Triangle Inequality Theorem, a fundamental principle in geometry that dictates the relationship between the sides of a triangle. 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. We applied this theorem diligently, setting up inequalities for all possible combinations of sides. By doing so, we discovered two crucial conditions for the third side, which we called "c". First, we found that c must be less than 18 (c < 18), because the sum of the other two sides (6 and 12) is 18. Second, we determined that c must be greater than 6 (c > 6), because the sum of the third side and the side with length 6 must be greater than the side with length 12. Putting these two conditions together, we arrived at a compound inequality: 6 < c < 18. This means that the length of the third side must fall strictly between 6 and 18 units. It cannot be exactly 6 units, and it cannot be exactly 18 units – it has to be somewhere in between. Then, we turned our attention to the given options and evaluated each one against our calculated range. We quickly dismissed options A and B because they either presented contradictory conditions or included negative values for side lengths, which are not possible in geometry. Finally, we landed on option C, which perfectly matched our range of 6 < c < 18. This confirmed that option C correctly identifies the possible side lengths for the third side of the triangle. So, to summarize, the Triangle Inequality Theorem is a powerful tool for understanding the relationships between the sides of a triangle, and we've successfully used it to solve our problem. We've not only found the correct answer but also gained a deeper understanding of how triangles work. Keep this theorem in your mathematical toolkit, and you'll be well-equipped to tackle similar problems in the future!