Triangle Inequality Theorem Find Possible Values Of N

Hey guys! Ever wondered what makes a triangle a triangle? It's not just any three lines slapped together, you know! There's a fundamental rule that governs the relationship between the sides, and it's called the Triangle Inequality Theorem. This theorem is super important in geometry, and it's surprisingly simple to understand. So, let's dive into a problem that puts this theorem to the test. We'll explore how to find the possible range of values for the third side of a triangle when we already know the lengths of the other two sides. Get ready to sharpen those geometry skills!

Understanding the Triangle Inequality Theorem

Before we tackle the problem, let's make sure we're all on the same page about the Triangle Inequality Theorem. This theorem is the key to solving our problem, and it's a concept you'll use again and again in geometry. Simply put, the 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's a seemingly simple rule, but it's incredibly powerful. Think about it – if two sides were shorter than the third side, they wouldn't be able to “reach” each other to form a closed triangle. Imagine trying to build a triangle with sticks of lengths 2 cm, 3 cm, and 10 cm. The 2 cm and 3 cm sticks simply wouldn't be long enough to connect and form a triangle with the 10 cm stick. The Triangle Inequality Theorem formalizes this intuitive idea. To fully grasp this, let's represent the sides of a triangle as a, b, and c. The Triangle Inequality Theorem gives us three inequalities that must all be true for the sides to form a triangle:

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

These three inequalities might seem like a lot to remember, but they all say the same thing – the sum of any two sides must be greater than the third side. By checking all three inequalities, we can confidently determine if three given side lengths can actually form a triangle. Furthermore, this theorem isn't just about checking if a triangle can exist; it also helps us determine the possible range of values for a missing side. This is exactly what we'll be doing in the problem we're about to solve.

Solving the Triangle Side Length Problem

Now, let's get to the main event! Here's the problem we're going to solve:

A triangle has side lengths measuring 20 cm, 5 cm, and n cm. Which inequality describes the possible values of n?

A. 5 extless n extless 15

B. 5 extless n extless 20

C. 15 extless n extless 20

D. 15 extless n extless 25

This is a classic problem that directly applies the Triangle Inequality Theorem. We're given two sides of a triangle (20 cm and 5 cm) and asked to find the possible range of values for the third side (n). To do this, we'll use the three inequalities from the Triangle Inequality Theorem. Let's treat the sides as follows: a = 20 cm, b = 5 cm, and c = n cm. Now, we can plug these values into our inequalities:

1. 20 + 5 extgreater n which simplifies to 25 extgreater n
2. 20 + n extgreater 5
3. 5 + n extgreater 20

The first inequality, 25 extgreater n, tells us that n must be less than 25. This makes sense – the third side can't be longer than the sum of the other two sides. Now, let's look at the second inequality: 20 + n extgreater 5. To isolate n, we subtract 20 from both sides, giving us n extgreater -15. While this is mathematically correct, it's not particularly helpful in our context. Side lengths can't be negative, so this inequality doesn't give us a meaningful lower bound for n. The third inequality, 5 + n extgreater 20, is where we'll find our lower bound. Subtracting 5 from both sides, we get n extgreater 15. This means that n must be greater than 15. If n were less than or equal to 15, the sides 5 cm and n cm wouldn't be long enough to “reach” and form a triangle with the 20 cm side.

Combining the Inequalities to Find the Range

Okay, so we've determined that n must be less than 25 (from 25 extgreater n) and greater than 15 (from n extgreater 15). Now, how do we combine these two pieces of information to get the possible range of values for n? This is where we express the solution as a compound inequality. We know that n is simultaneously greater than 15 AND less than 25. We can write this as: 15 extless n extless 25

This compound inequality tells us that n can be any value between 15 cm and 25 cm (but not equal to 15 cm or 25 cm). If n were exactly 15 cm, the sides 5 cm, 15 cm, and 20 cm wouldn't form a triangle (5 + 15 = 20, violating the Triangle Inequality Theorem). Similarly, if n were exactly 25 cm, the sides 20 cm, 5 cm, and 25 cm wouldn't form a triangle (20 + 5 = 25, again violating the theorem). Therefore, n must lie strictly between 15 cm and 25 cm. Looking back at our answer choices, we can see that the correct answer is:

D. 15 extless n extless 25

Key Takeaways and Tips

Great job, guys! You've successfully navigated the Triangle Inequality Theorem and found the possible range for the third side of a triangle. Let's recap the key steps and highlight some tips for tackling similar problems in the future:

• Remember the Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• Write out all three inequalities: Don't just focus on one pair of sides; make sure to consider all three combinations to get a complete picture.
• Solve each inequality: Isolate the unknown variable (n in our case) to find the individual constraints.
• Combine the inequalities: Express the solution as a compound inequality to represent the range of possible values.
• Consider the context: Remember that side lengths can't be negative, so any negative solutions can be disregarded.
• Check the endpoints: Be mindful of whether the endpoints of the range are included or excluded. In our problem, n had to be strictly between 15 and 25, so we used “ extless” symbols.

By following these steps and keeping the Triangle Inequality Theorem in mind, you'll be well-equipped to handle a wide variety of triangle problems. Practice makes perfect, so try working through some more examples to solidify your understanding. Geometry can be a lot of fun, especially when you have the right tools and techniques at your disposal!

Practice Problems to Sharpen Your Skills

To really master the Triangle Inequality Theorem, it's essential to practice applying it in different scenarios. Here are a few problems you can try to test your understanding:

1. Two sides of a triangle measure 8 cm and 12 cm. What are the possible values for the length of the third side?
2. Can a triangle be formed with sides of lengths 4 cm, 5 cm, and 9 cm? Why or why not?
3. The lengths of two sides of a triangle are 10 inches and 15 inches. If the length of the third side is an integer, what is the largest possible length of the third side?
4. A triangle has sides of length x, x + 3, and 2x. Find the possible values of x.

Working through these problems will help you build confidence in using the Triangle Inequality Theorem and develop your problem-solving skills. Don't be afraid to draw diagrams and visualize the triangles – this can often make the relationships between the sides clearer. Remember to always check your answers and make sure they make sense in the context of the problem.

By tackling these practice problems and consistently reviewing the key concepts, you'll become a true geometry whiz in no time! Keep up the great work, guys, and happy problem-solving!