Determining The Range Of The Third Side In An Acute Triangle

Hey guys! Today, we're diving deep into the fascinating world of acute triangles, specifically focusing on how to determine the possible range of values for the third side when we already know the lengths of the other two. This is a classic geometry problem that often pops up in math competitions and exams, so let's break it down in a way that's super easy to understand.

Understanding the Triangle Inequality Theorem

Before we jump into the specifics of acute triangles, it's crucial to grasp the Triangle Inequality Theorem. This theorem is the cornerstone of solving any triangle-related problem, and it's surprisingly simple. 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. Think of it this way: if you have two short sticks, they can't possibly reach each other to form a triangle if the third stick is too long. This fundamental concept helps us establish the initial boundaries for the possible range of the third side, s. In our case, we have two sides measuring 8 cm and 10 cm. Applying the Triangle Inequality Theorem, we get three key inequalities:

1. 8 + 10 > s
2. 8 + s > 10
3. 10 + s > 8

The first inequality, 8 + 10 > s, simplifies to 18 > s. This tells us that the third side, s, must be less than 18 cm. The second inequality, 8 + s > 10, rearranges to s > 2. This indicates that the third side must be greater than 2 cm. The third inequality, 10 + s > 8, gives us s > -2, which is always true since side lengths cannot be negative. So, from the Triangle Inequality Theorem, we've established a preliminary range: 2 < s < 18. But hold on, we're not done yet! This range only ensures that we can form any triangle, not necessarily an acute triangle. To ensure it is an acute triangle, we have to consider additional criteria.

The Acute Triangle Condition

Now comes the exciting part – figuring out what makes a triangle acute. An acute triangle is defined as a triangle where all three angles are less than 90 degrees. To determine if a triangle is acute, we use a modified version of the Pythagorean Theorem. Remember the classic Pythagorean Theorem, a² + b² = c², which applies to right triangles? For acute triangles, the relationship changes slightly. If 'c' is the longest side, then for an acute triangle, we need a² + b² > c². This condition ensures that the angle opposite the longest side is less than 90 degrees. This is a critical step in pinpointing the range for s. Let's apply this to our problem. We have sides of 8 cm, 10 cm, and s. We need to consider two cases:

Case 1: 10 cm is the longest side

If 10 cm is the longest side, then s must be less than 10 cm. Our acute triangle condition becomes:

8² + s² > 10²

64 + s² > 100

s² > 36

s > 6

So, in this case, s must be greater than 6 cm.

Case 2: s is the longest side

If s is the longest side, then s must be greater than 10 cm. Our acute triangle condition now looks like this:

8² + 10² > s²

64 + 100 > s²

164 > s²

s < √164

s < 12.81 (approximately)

This tells us that if s is the longest side, it must be less than approximately 12.81 cm.

Combining the Conditions

Alright, guys, we've done the heavy lifting! Now, let's put all the pieces together. We have several conditions for the possible range of s:

• From the Triangle Inequality Theorem: 2 < s < 18
• From the acute triangle condition (Case 1): s > 6 (when 10 cm is the longest side)
• From the acute triangle condition (Case 2): s < √164 ≈ 12.81 (when s is the longest side)

To find the best representation of the possible range of values for s, we need to consider the intersection of these conditions. The most restrictive lower bound is 6 cm, and the most restrictive upper bound is approximately 12.81 cm. Therefore, the final range for s is:

6 < s < 12.81 (approximately)

This means that the third side, s, must be greater than 6 cm and less than approximately 12.81 cm to form an acute triangle with sides of 8 cm and 10 cm.

Why is this important?

You might be wondering, why do we even need to know this? Well, understanding the relationships between side lengths and angles in triangles is fundamental to many areas of mathematics and real-world applications. From architecture and engineering to navigation and computer graphics, triangles are everywhere! Being able to determine the possible ranges of side lengths helps us ensure the stability and functionality of structures, calculate distances accurately, and create realistic 3D models. It is important to ensure we have accurate data and measurements when dealing with triangles in any practical context.

Common Mistakes to Avoid

Now, let's talk about some common pitfalls that students often encounter when tackling these types of problems:

• Forgetting the Triangle Inequality Theorem: This is the most common mistake. Always start by applying the Triangle Inequality Theorem to establish the initial range for the third side.
• Only considering one case for the acute triangle condition: Remember, you need to consider both cases – when the given side (in this case, 10 cm) is the longest and when the unknown side (s) is the longest.
• Confusing acute and obtuse triangle conditions: Make sure you use the correct inequality (a² + b² > c² for acute and a² + b² < c² for obtuse).
• Not simplifying the inequalities: Always simplify the inequalities to get the most accurate range for s.

Avoiding these mistakes will help you solve these problems with confidence and precision.

Let's Practice!

To solidify your understanding, let's try a similar problem. Suppose you have a triangle with sides measuring 5 cm and 7 cm. What is the possible range of values for the third side, s, if the triangle is acute? Try solving this problem using the steps we discussed, and feel free to share your answer in the comments below!

Conclusion

So there you have it! Determining the possible range of values for the third side of an acute triangle involves a combination of the Triangle Inequality Theorem and the acute triangle condition. By carefully considering these principles and avoiding common mistakes, you can master these types of problems and gain a deeper appreciation for the beautiful world of geometry. Keep practicing, and you'll be a triangle expert in no time!