Finding The Range Of A Triangle's Third Side

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Hey math enthusiasts! Today, we're diving into a classic geometry problem: determining the possible lengths of the third side of a triangle, given the lengths of the other two sides. This is super important because it helps us understand the fundamental rules that govern triangles. Let's break it down. In this scenario, we're given a triangle with side lengths of 2x + 2 ft, x + 3 ft, and n ft. The question asks us to find the range of possible values for n. This is where the Triangle Inequality Theorem comes into play. This theorem is the golden rule for triangles, and it dictates that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this rule isn't followed, you simply can't form a closed triangle. It's like trying to build a house with walls that are too short – it just won't stand up!

So, how do we use this theorem to find the possible values of n? Easy peasy! We'll apply the theorem to all three possible combinations of sides and create inequalities. This will give us the boundaries within which n must lie. Then, we will use the inequalities we create to find a suitable range. First, let's consider the case where the sides 2x + 2 and x + 3 are added together. According to the Triangle Inequality Theorem, this sum must be greater than n: (2x + 2) + (x + 3) > n. Simplifying this, we get 3x + 5 > n. Next, let's think about the other two combinations. For the next combination, the sides 2x + 2 and n must add up to be greater than x + 3: (2x + 2) + n > x + 3. If we simplify this, we get n > x + 1. Finally, for the last possible combination, we need x + 3 and n to be greater than 2x + 2: (x + 3) + n > 2x + 2, which simplifies to n > x - 1. The three inequalities we got are 3x + 5 > n, n > x + 1, and n > x - 1. It is important to realize that, because n must be less than 3x + 5, n must be lower than the value of 3x + 5, but greater than x + 1. This gives us the possible range for n! If we think about it, this is quite logical, right? The third side has to be long enough to complete the triangle but not so long that it becomes an open figure. It has to be within a particular range determined by the other two sides.

Breaking Down the Triangle Inequality Theorem

Alright, let's get into the nitty-gritty of the Triangle Inequality Theorem. As we mentioned earlier, this theorem is the cornerstone of understanding the relationship between the sides of a triangle. In essence, it states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is because triangles are closed figures. A triangle can only be created if its sides can meet at the vertices. If the sum of two sides is less than or equal to the third side, the two sides cannot meet, and thus, a triangle cannot be formed. This simple yet powerful concept has vast implications in geometry and other areas of mathematics. Let's consider a few examples to drive home the point. Suppose we have a triangle with sides 3, 4, and 5. Applying the theorem, we have:

  • 3 + 4 > 5 (True)
  • 3 + 5 > 4 (True)
  • 4 + 5 > 3 (True)

Since all conditions are met, a triangle can be formed with these side lengths. On the other hand, let's try a combination that violates the theorem, such as sides 1, 2, and 4. Applying the theorem:

  • 1 + 2 > 4 (False)

Since the first inequality is already false, we know that a triangle cannot be formed with sides 1, 2, and 4. This example emphasizes the importance of the theorem in determining the feasibility of a triangle. Now, what happens if we have sides 6, 8, and 10? This is also a valid triangle! Applying the theorem:

  • 6 + 8 > 10 (True)
  • 6 + 10 > 8 (True)
  • 8 + 10 > 6 (True)

So, as you can see, the theorem holds true for all types of triangles, whether it's a scalene, isosceles, or equilateral triangle. It does not matter. The key is that the sum of any two sides must exceed the third. Remember, the Triangle Inequality Theorem is our guiding principle in problems like the one we're discussing. It helps us constrain the possible values for the third side and solve a variety of geometry problems.

Applying the Theorem Step-by-Step

Let's walk through another example. Suppose we're given two sides of a triangle with lengths 7 and 10. What's the range of possible values for the third side, let's call it s? The first thing we do is apply the theorem. We know that the sum of any two sides must be greater than the third. This gives us three inequalities:

  1. 7 + 10 > s => 17 > s (or s < 17)
  2. 7 + s > 10 => s > 3
  3. 10 + s > 7 => s > -3

Now, looking at the result, we know that the third side, s, has to be less than 17 and greater than 3. The third inequality tells us that s has to be greater than -3, but since a side cannot have a negative length, we can ignore it. Thus, combining the inequalities, we find that 3 < s < 17. This means that the third side must be greater than 3 but less than 17. Any value within this range is a valid length for the third side. Let's try one more example. Let's say we have sides with lengths 5 and 12. What is the range for t? Once again, we use the Triangle Inequality Theorem. So, we know that:

  1. 5 + 12 > t => 17 > t (or t < 17)
  2. 5 + t > 12 => t > 7
  3. 12 + t > 5 => t > -7

We can combine the inequalities, so 7 < t < 17. This means that the third side has to be greater than 7 but less than 17. If we have any value within this range, it will make a valid triangle. Understanding and applying the Triangle Inequality Theorem is critical for these problems. It helps us define the boundaries of possible side lengths, ensuring that our triangles can actually exist in the world of geometry. So, keep this theorem in mind, and you'll be well on your way to acing these kinds of problems.

Conclusion: Finding the Correct Expression

Back to our original question! We want to find the correct expression that represents the possible values of n. Earlier, we found three inequalities, which were 3x + 5 > n, n > x + 1, and n > x - 1. These inequalities define the upper and lower bounds for n. Let's evaluate the options. In option A, we have x - 1. This expression is a possible value for n. We know, based on our calculations, that n must be greater than x - 1, but this does not mean that it's a valid expression for the range of values for n. Option B states that n = x - 1. This can't be correct since n must be greater than x - 1, meaning it is a limit. So, this is not the right answer either. In option C, we have 3x + 5. We know that n must be less than 3x + 5. So, this is also not correct since it states that it is equal to n. Finally, let's consider option D, which states that n = 3x + 5. Since n must be less than 3x + 5, this statement is also not correct. By analyzing the inequalities we got from the Triangle Inequality Theorem, we can realize that option C is the correct answer. While option C is not a range, the theorem says that n has to be less than 3x + 5. Although it does not give us a concrete range, it gives us the upper bound, which is what the question is asking us to do. Since the answer asks us what n could be, 3x + 5 fits the requirement. So, the correct answer is C. This question demonstrates how the Triangle Inequality Theorem helps us establish the range of possible side lengths in a triangle. By understanding and applying the theorem, we can accurately solve for unknown side lengths and deepen our understanding of geometric principles. Keep practicing, and you'll become a triangle master in no time!