Triangle Inequality Theorem Explained Finding Possible Side Lengths

Hey guys! Ever wondered if any three lengths can form a triangle? It's a fascinating question that dives into the heart of geometry. Today, we're going to tackle a problem that Chang is facing: figuring out the possible side lengths of a triangle when one side is already known. This involves a crucial concept called the Triangle Inequality Theorem. So, grab your thinking caps, and let's dive in!

The Triangle Inequality Theorem: The Golden Rule of Triangles

At the core of this problem lies the Triangle Inequality Theorem, which is like the golden rule for triangles. 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. Sounds a bit complicated? Let's break it down.

Imagine you have three sticks. To form a triangle, you need to be able to connect the ends of these sticks. If two of the sticks are too short compared to the third, they won't be able to reach each other to form a closed shape. The Triangle Inequality Theorem puts this idea into mathematical terms. It's not enough for the sides to just exist; they have to be the right lengths to actually create a triangle. This theorem isn't just some abstract rule; it's a fundamental principle that governs the very existence of triangles. Without it, our understanding of geometry would be vastly different, and many of the structures and calculations we rely on in fields like engineering and architecture wouldn't be possible. So, understanding this theorem is like unlocking a secret code to the world of triangles.

Think of it this way: if you have sides a, b, and c, then the following three conditions must be true:

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

All three of these conditions must be met for the sides to form a triangle. If even one of these inequalities doesn't hold, then you can't make a triangle with those side lengths. This is super important, so let's keep it in mind as we work through Chang's problem. This might seem like a lot to remember, but it becomes intuitive with practice. Imagine trying to build a triangle with flimsy sticks – you quickly realize that the shorter sticks need to combine to be longer than the longest stick to even have a chance of connecting. The theorem just puts this real-world experience into precise mathematical language, making it a powerful tool for solving problems like Chang's.

Chang's Triangle Challenge: Applying the Theorem

Now, let's get back to Chang's problem. We know one side of the triangle is 13 cm. The question asks us to find a set of two sides that could be the other two sides of this triangle. We're given four options:

A. 5 cm and 8 cm B. 6 cm and 7 cm C. 7 cm and 2 cm D. 8 cm and 9 cm

To solve this, we'll use the Triangle Inequality Theorem. For each option, we need to check if all three conditions of the theorem are met. This means we'll need to do a little bit of adding and comparing. But don't worry, it's not as complicated as it sounds! We're essentially just checking if the two shorter sides combined are longer than the longest side in each case. If they are, then we're one step closer to finding the right answer. If not, we can immediately rule out that option and move on to the next. This process of elimination is a powerful problem-solving technique in math, and it's exactly what we'll be using here.

Let's go through each option systematically and see which one fits the bill. Remember, we're not just looking for any combination of sides; we're looking for the only combination that adheres to the strict rules of the Triangle Inequality Theorem. This highlights the importance of precision in math – a single wrong calculation or missed inequality can lead us to the wrong answer. So, let's be careful, methodical, and make sure we're applying the theorem correctly in each case.

Option A: 5 cm and 8 cm – A No-Go

Let's start with option A: 5 cm and 8 cm. We have the three sides: 5 cm, 8 cm, and 13 cm. Now, we need to check if the Triangle Inequality Theorem holds true. Remember, all three conditions must be met for these sides to form a triangle.

• 5 + 8 > 13 ? This simplifies to 13 > 13, which is false.

Since the first condition itself is not met, we don't even need to check the other two. The sum of 5 cm and 8 cm is equal to 13 cm, not greater than 13 cm. This means that these sides cannot form a triangle. Imagine trying to build a triangle with these lengths – the two shorter sides would lie flat against the longest side, never meeting to form a closed shape. This vividly illustrates why the Triangle Inequality Theorem is so important – it ensures that the sides are long enough to actually connect and create a triangle.

Therefore, option A is incorrect. We can confidently cross this off our list and move on to the next option, knowing that it doesn't satisfy the fundamental requirements for a triangle to exist. This is a great example of how the Triangle Inequality Theorem can quickly eliminate possibilities and narrow down our search for the correct answer.

Option B: 6 cm and 7 cm – Another Disqualification

Now, let's consider option B: 6 cm and 7 cm. Our sides are now 6 cm, 7 cm, and 13 cm. Let's put them to the test using the Triangle Inequality Theorem.

• 6 + 7 > 13 ? This simplifies to 13 > 13, which is again false.

Just like in option A, the sum of the two shorter sides (6 cm and 7 cm) is equal to the longest side (13 cm), not greater than it. This means option B also fails the Triangle Inequality Theorem. We're seeing a pattern here, aren't we? It's not enough for the sides to just be close in length; they have to have a significant enough difference to allow for a true triangle to be formed. Think about it – if the two shorter sides barely reach the length of the longest side, there's no way they can bend and connect to form the other two sides of the triangle. They would simply lie flat against the longest side, creating a straight line instead of a closed shape.

So, option B is also incorrect. We're making progress though! By systematically testing each option, we're getting closer to finding the one that works. Let's keep going and see what option C has in store for us.

Option C: 7 cm and 2 cm – A Definite No

Let's move on to option C: 7 cm and 2 cm. This gives us side lengths of 7 cm, 2 cm, and 13 cm. Time to apply the Triangle Inequality Theorem!

• 7 + 2 > 13 ? This simplifies to 9 > 13, which is definitely false.

In this case, the sum of 7 cm and 2 cm (which is 9 cm) is less than 13 cm. This means that these side lengths cannot possibly form a triangle. This is even more evident than the previous examples – the two shorter sides are significantly shorter than the longest side, making it impossible for them to connect and form a closed shape. Imagine trying to build a triangle with one very long stick and two very short sticks – the short sticks simply wouldn't be able to reach each other, no matter how you arrange them.

Option C is incorrect. We've eliminated three options already! This is fantastic – it means that if option D passes the test, it has to be the correct answer. Let's go for it!

Option D: 8 cm and 9 cm – The Winner!

Finally, let's examine option D: 8 cm and 9 cm. Our side lengths are 8 cm, 9 cm, and 13 cm. Let's check the Triangle Inequality Theorem one last time.

• 8 + 9 > 13 ? This simplifies to 17 > 13, which is true.
• 8 + 13 > 9 ? This simplifies to 21 > 9, which is true.
• 9 + 13 > 8 ? This simplifies to 22 > 8, which is true.

All three conditions of the Triangle Inequality Theorem are met! This means that the sides 8 cm, 9 cm, and 13 cm can indeed form a triangle. We've found our answer!

Option D is the correct answer. This means that Chang can have a triangle with sides of 13 cm, 8 cm, and 9 cm. This highlights the power of the Triangle Inequality Theorem – it allows us to definitively determine whether a set of side lengths can form a triangle, and in this case, it led us to the correct answer after systematically eliminating the other possibilities. It's like a mathematical detective, helping us solve the mystery of which sides can create a triangle!

Conclusion: Mastering the Triangle Inequality Theorem

So, by using the Triangle Inequality Theorem, we were able to help Chang find the possible side lengths for his triangle. The key takeaway here is that the sum of any two sides of a triangle must be greater than the third side. This theorem is a fundamental concept in geometry, and understanding it can help you solve a variety of problems related to triangles.

Remember guys, math isn't just about memorizing formulas; it's about understanding the underlying principles and applying them to real-world problems. The Triangle Inequality Theorem is a perfect example of this – it's a simple concept that has powerful implications in geometry and beyond. Keep practicing, keep exploring, and you'll master these concepts in no time! And now you know, not just any three lengths can make a triangle – there's a rule for that!