Hey everyone! Today, we're diving into a fascinating concept in geometry: the Triangle Inequality Theorem. This theorem is super useful for figuring out the possible lengths of sides in a triangle. We'll be tackling a problem where we're given two side lengths and need to determine the range of possible values for the third side. So, let's jump right in and get our geometry muscles flexing!
Understanding the Triangle Inequality Theorem
Before we tackle the problem directly, let's make sure we've got a solid grasp of the Triangle Inequality Theorem. In simple terms, 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. It sounds a bit like a mouthful, but it's actually quite intuitive once you visualize it. Think about trying to build a triangle with toothpicks. If you have two short toothpicks and one very long one, you won't be able to connect them to form a triangle – the two shorter ones just won't reach! This is the essence of the Triangle Inequality Theorem in action. To put it mathematically, if we have a triangle with side lengths a, b, and c, the following three inequalities must hold true:
- a + b > c
- a + c > b
- b + c > a
These three inequalities are the key to solving problems involving possible side lengths. They ensure that the triangle can actually "close" and form a valid shape. Understanding these inequalities is crucial, guys, so let's break them down further. Imagine side c is the longest side. If the sum of sides a and b is only equal to side c, then the two shorter sides will just lie flat along side c, and you won't have a triangle – just a straight line. If the sum of a and b is less than c, then they definitely won't be able to reach each other to form a closed shape. Only when a + b is greater than c can we actually form a triangle. The same logic applies to the other two inequalities, ensuring that no single side is too long relative to the other two. So, with this theorem in our toolkit, we're ready to tackle the problem at hand! Remember, it's all about making sure those sides can connect and form a triangle.
Applying the Theorem to Our Problem
Okay, now let's apply the Triangle Inequality Theorem to the problem we're facing. We've got a triangle with sides of 20 cm, 5 cm, and n cm. Our mission is to figure out the possible values of n. To do this, we'll use the three inequalities we discussed earlier. We'll substitute the given side lengths into the inequalities and see what constraints we can find on n. This is where the math gets really practical, guys! We're not just learning a theorem in abstract; we're actually using it to solve a concrete problem. This is the beauty of geometry – it connects theoretical ideas to real-world situations. Let's start by plugging the side lengths into our inequalities:
- 20 + 5 > n => 25 > n
- 20 + n > 5
- 5 + n > 20
The first inequality, 25 > n, tells us that n must be less than 25. This makes sense – if n were greater than or equal to 25, then the sum of the other two sides (20 and 5) wouldn't be greater than n, and we couldn't form a triangle. Now let's look at the second inequality, 20 + n > 5. If we subtract 20 from both sides, we get n > -15. This might seem a bit strange at first, but remember that side lengths can't be negative. So, this inequality doesn't actually give us a useful lower bound for n. It's essentially telling us something we already know: n has to be a positive number. The real meat of the problem comes from the third inequality: 5 + n > 20. If we subtract 5 from both sides, we get n > 15. This is a crucial piece of information! It tells us that n must be greater than 15. If n were less than or equal to 15, then the sum of 5 and n wouldn't be greater than 20, and again, we couldn't form a triangle. So, now we have two key pieces of information: n must be less than 25, and n must be greater than 15. This gives us a range of possible values for n. Let's put it all together and see what our final answer is!
Determining the Possible Values of n
Alright, we've done the hard work of applying the Triangle Inequality Theorem and setting up our inequalities. Now, let's bring it all home and figure out the possible values of n. We've established that n must satisfy two conditions:
- n < 25
- n > 15
Combining these two inequalities, we get 15 < n < 25. This means that n can be any value between 15 and 25, but it can't be equal to 15 or 25. Think of it like a sweet spot for the side length n. If it's too small (less than or equal to 15), then the 5 cm and n cm sides won't be long enough to meet the 20 cm side. If it's too big (greater than or equal to 25), then the 20 cm and 5 cm sides won't be long enough to meet the n cm side. Only values within this range will allow us to form a valid triangle. So, the possible values of n are all the numbers between 15 and 25. Now, let's take a look at the answer choices provided in the original problem and see which one matches our solution. This is where we connect our mathematical work to the specific options given. Sometimes, problems like these are designed to have tricky answer choices, so it's crucial to make sure we're carefully comparing our solution to the options. We want to make sure we're not just picking the first answer that looks close; we want to be absolutely sure it's the correct one. So, let's put on our detective hats and compare our range of 15 < n < 25 to the answer choices.
Looking at the provided options, we see the following:
A. 5 < n < 15 B. 5 < n < 20
Neither of these options matches our solution of 15 < n < 25. Therefore, there seems to be an error in the provided answer choices. The correct answer should reflect the range 15 < n < 25. It's always a good idea to double-check your work when you encounter a situation like this, just to make sure you haven't made any mistakes. But in this case, we've carefully applied the Triangle Inequality Theorem and arrived at a clear solution. So, the most likely explanation is that the answer choices themselves contain an error. This highlights the importance of understanding the underlying concepts and not just relying on memorization or pattern matching. Even if the answer choices don't perfectly align with your solution, you can still be confident in your answer if you've followed the correct logical steps.
Conclusion: Mastering the Triangle Inequality Theorem
So, there you have it! We've successfully navigated the Triangle Inequality Theorem and figured out the possible values for the side length n. We saw how the theorem helps us determine if three given side lengths can actually form a triangle. This is a fundamental concept in geometry, and it's super useful in various applications, from construction to navigation. Guys, mastering this theorem is a great step toward building a strong foundation in geometry. Remember, the key is to understand that the sum of any two sides of a triangle must be greater than the third side. Keep practicing with different examples, and you'll become a Triangle Inequality Theorem pro in no time! And don't be afraid to question the answer choices if they don't seem right – sometimes, even the problems themselves can have errors. The most important thing is to understand the math and be confident in your reasoning.