Finding The Third Side: Acute Triangle Solutions

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Finding the Third Side: Acute Triangle Solutions

Hey guys! Let's dive into a classic geometry problem that's all about acute triangles. We're given a triangle with two sides measuring 8 cm and 10 cm, and the question is: What's the best way to represent the possible range of values for the third side, which we'll call s? This isn't just about finding a value; it's about understanding the limits within which that third side can exist to still form a valid, and in this case, acute triangle. This problem combines the basic triangle inequality theorem with the specific properties of acute triangles, making it a bit more involved, but super interesting! So, let's break it down step-by-step to make sure we nail it. Understanding this will give you a solid foundation for tackling other geometry problems, especially those involving triangles and their properties. We will first look at how to find the range of the third side by applying the triangle inequality theorem. Afterward, we'll refine the range using the acute triangle condition. So let's get started.

Understanding the Basics: Triangle Inequality Theorem

Alright, first things first, let's talk about the Triangle Inequality Theorem. This is the golden rule when it comes to figuring out if three side lengths can even form a triangle in the first place. 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. This ensures that the sides can actually connect to form a closed shape. It’s like saying, "Hey, if you want to build a triangle, each side needs to be shorter than the combined length of the other two sides."

So, with our sides of 8 cm and 10 cm, let’s consider how the third side, s, has to behave.

  1. 8 cm + 10 cm > s => 18 cm > s or s < 18 cm. This means the third side must be less than 18 cm.
  2. 8 cm + s > 10 cm => s > 2 cm. This tells us the third side must be greater than 2 cm.
  3. 10 cm + s > 8 cm. => s > -2 cm. Since side lengths cannot be negative, this is always true when s>0.

Combining these, we get 2 cm < s < 18 cm. This is the initial range of possible values for s based on the Triangle Inequality Theorem. Any value outside this range simply won't form a triangle. But, keep in mind, we're not just looking for any triangle; we're after an acute triangle, which is a special type of triangle where all interior angles are less than 90 degrees. This extra condition will change the range we just found. It helps to visualize this; imagine the third side getting longer and shorter. There's a sweet spot in the middle where the triangle is acute, but as the sides change, we cross over to obtuse triangles or, in the limiting cases, straight lines that don't make triangles. Therefore, the triangle inequality theorem gives us a solid starting point, but it's not the whole story.

Diving Deeper: Acute Triangle Properties

Now, let's shift gears and consider what makes a triangle acute. In an acute triangle, all three angles are less than 90 degrees. This is where things get a bit more interesting, because this condition gives us a more refined range for the third side. The property we use here is related to the Pythagorean theorem, which you guys probably already know. For a right triangle, a² + b² = c², where c is the longest side (the hypotenuse). For an acute triangle, the square of the longest side must be less than the sum of the squares of the other two sides. This is because, as the angles open up, the longest side's length decreases. In our case, the longest side will be s, so we need to consider two scenarios:

  1. If s is the longest side (i.e., s > 10 cm), then we need: 8² + 10² > s² 64 + 100 > s² 164 > s² √164 > s => s < √164 ≈ 12.8 cm
  2. If 10 cm is the longest side (i.e., s < 10 cm), then: s² + 8² > 10² s² + 64 > 100 s² > 36 s > 6 cm

Therefore, considering both conditions for an acute triangle: if s is the longest side, then s < 12.8 cm (from the first condition), and if 10 cm is the longest side, then s > 6 cm (from the second condition). This means the range of values for an acute triangle is 6 cm < s < 12.8 cm. This is a much smaller range than the initial 2 cm < s < 18 cm given by the triangle inequality theorem. This highlights the importance of incorporating the acute triangle condition into our solution. It’s like adding another layer of requirements to the problem, refining our options, and giving us a more precise answer. Remember, the triangle must satisfy the triangle inequality theorem and the acute triangle condition. This ensures that the third side not only forms a valid triangle but also that all its angles are less than 90 degrees. Understanding this concept is crucial for any geometry enthusiast. Now let's compare these calculations with the options to select the correct answer.

Finding the Correct Answer

Now that we've crunched the numbers, let's look at the multiple-choice options and see which one aligns with our findings. We've determined that for an acute triangle with sides of 8 cm and 10 cm, the third side, s, must satisfy the conditions 6 cm < s < 12.8 cm. The options provided are:

A. s < 2 B. s > 6 C. s < 2 or s > 18 D. s < 6 or s > 12.8

Based on our calculations:

  • Option A is incorrect because the third side must be greater than 6 cm and can't be less than 2 cm.
  • Option B is partially correct since the third side must be greater than 6 cm. This doesn't consider the upper limit of 12.8 cm, therefore it is not the complete answer.
  • Option C is incorrect because the third side must be between 6 and 12.8 and not less than 2 or greater than 18. This is the condition defined by the triangle inequality theorem and excludes the acute triangle condition.
  • Option D is the most accurate representation of the values. It correctly suggests the exclusion range for s to be less than 6 cm or greater than 12.8 cm. This is because the triangle is not acute if s falls outside of the interval (6, 12.8). Therefore, the correct answer is D.

So, the answer is D: s < 6 or s > 12.8. Remember, understanding how to apply both the Triangle Inequality Theorem and the properties of acute triangles is key to solving this type of problem. Good job, guys! This problem isn’t just about memorizing formulas; it's about understanding the relationships between the sides and angles of triangles and applying the correct theorems to solve the problem. Practice makes perfect, so keep working through these types of problems, and you'll become a geometry whiz in no time. Keep the spirit of exploration and learning! Remember that mathematics is all about discovery, and with each problem you solve, you're gaining a deeper appreciation of the patterns and relationships that govern the world around us.