Obtuse Triangles: Which Sides Make The Cut?

by Andrew McMorgan 44 views

Hey guys! Today, we're diving deep into the world of triangles, specifically focusing on those obtuse triangles. You know, the ones that have that one really wide angle, greater than 90 degrees. We're going to figure out which sets of three numbers can actually form the sides of such a triangle. This isn't just about memorizing formulas; it's about understanding the why behind it. So, grab your notebooks (or just your brains!), and let's get this mathematical party started!

Understanding the Triangle Inequality Theorem

Before we can even think about obtuse triangles, we need to nail down a fundamental rule: the Triangle Inequality Theorem. This theorem is like the bouncer at a club – it decides who gets in and who doesn't. For any three lengths to form a triangle, the sum of any two sides must be greater than the third side. Seriously, guys, if this rule isn't met, you can forget about forming a triangle. It's that simple. Let's break it down with an example. If you have sides a, b, and c, then all three of these must be true: a + b > c, a + c > b, and b + c > a. If even one of these conditions fails, then those lengths won't connect to form a triangle. Think about trying to connect three sticks. If two of them are super short, and the third one is way too long, you just can't make them meet at the ends. That's the Triangle Inequality Theorem in action. It's a crucial first step, and we must check it for every single option before we even consider if it's obtuse or not. So, keep this theorem in your back pocket; it's going to be our trusty sidekick throughout this whole process. We'll be using it to eliminate any sets of numbers that just can't form a triangle at all.

The Pythagorean Theorem and Its Cousins: Acute, Right, and Obtuse

Now, let's talk about the Pythagorean Theorem. Most of you probably know it as a² + b² = c², which specifically applies to right triangles, where c is the hypotenuse (the longest side). But this theorem has some really cool relatives that help us classify triangles based on their angles. These relatives are what tell us if a triangle is acute, right, or obtuse. Let a and b be the lengths of the two shorter sides, and c be the length of the longest side. We compare a² + b² to c²:

  • If a² + b² > c², the triangle is acute. This means all three angles are less than 90 degrees. Think of a nicely proportioned triangle, not too stretched out.
  • If a² + b² = c², the triangle is right. This is the classic Pythagorean theorem case, where you have one perfect 90-degree angle.
  • If a² + b² < c², the triangle is obtuse. This is the one we're after, guys! This inequality means that the angle opposite the longest side (c) is greater than 90 degrees. The triangle is stretched out, making one angle feel particularly large.

This comparison is the key to identifying our obtuse triangles. It's like a secret code that unlocks the type of triangle we're dealing with. Remember, c always has to be the longest side for these comparisons to work. If you mess that up, the whole system falls apart. So, make sure you identify the longest side first, then plug the numbers into the a² + b² vs. c² equation. This is where the magic happens and we can finally pinpoint those obtuse triangles.

Let's Analyze the Options!

Alright, mathletes, it's time to put our knowledge to the test! We've got five sets of numbers, and we need to figure out which ones represent the sides of an obtuse triangle. Remember our two main rules:

  1. Triangle Inequality Theorem: The sum of any two sides must be greater than the third side.
  2. Obtuse Angle Test: For the longest side c, a² + b² < c².

Let's go through each option, one by one. No skipping steps, folks!

Option A: 4, 7, 8

First, let's check the Triangle Inequality Theorem. The longest side is 8.

  • Is 4 + 7 > 8? Yes, 11 > 8.
  • Is 4 + 8 > 7? Yes, 12 > 7.
  • Is 7 + 8 > 4? Yes, 15 > 4.

Great! These lengths can form a triangle. Now, let's apply the Obtuse Angle Test. Our longest side c is 8. So, a = 4 and b = 7.

  • Calculate a² + b²: 4² + 7² = 16 + 49 = 65.
  • Calculate c²: 8² = 64.
  • Compare: Is a² + b² < c²? Is 65 < 64? No, it's not. In fact, 65 > 64, which means this triangle is acute. So, Option A is not an obtuse triangle. Bummer!

Option B: 3, 4, 5

Let's check the Triangle Inequality Theorem. The longest side is 5.

  • Is 3 + 4 > 5? Yes, 7 > 5.
  • Is 3 + 5 > 4? Yes, 8 > 4.
  • Is 4 + 5 > 3? Yes, 9 > 3.

Triangle Inequality holds! Now for the Obtuse Angle Test. Longest side c is 5. So, a = 3 and b = 4.

  • Calculate a² + b²: 3² + 4² = 9 + 16 = 25.
  • Calculate c²: 5² = 25.
  • Compare: Is a² + b² < c²? Is 25 < 25? No. Since a² + b² = c², this is a right triangle. Classic 3-4-5, guys! So, Option B is also not an obtuse triangle.

Option C: 2, 2, 3

First up, the Triangle Inequality Theorem. The longest side is 3.

  • Is 2 + 2 > 3? Yes, 4 > 3.
  • Is 2 + 3 > 2? Yes, 5 > 2.
  • Is 2 + 3 > 2? Yes, 5 > 2.

Good news, these sides can form a triangle! Now, let's apply the Obtuse Angle Test. Our longest side c is 3. So, a = 2 and b = 2.

  • Calculate a² + b²: 2² + 2² = 4 + 4 = 8.
  • Calculate c²: 3² = 9.
  • Compare: Is a² + b² < c²? Is 8 < 9? YES! Since 8 < 9, this triangle is obtuse. Awesome! Option C is a winner! This means the angle opposite the side of length 3 is greater than 90 degrees. Pretty cool, right?

Option D: 6, 8, 9

Let's tackle the Triangle Inequality Theorem. The longest side is 9.

  • Is 6 + 8 > 9? Yes, 14 > 9.
  • Is 6 + 9 > 8? Yes, 15 > 8.
  • Is 8 + 9 > 6? Yes, 17 > 6.

Triangle Inequality is satisfied. Now for the Obtuse Angle Test. Longest side c is 9. So, a = 6 and b = 8.

  • Calculate a² + b²: 6² + 8² = 36 + 64 = 100.
  • Calculate c²: 9² = 81.
  • Compare: Is a² + b² < c²? Is 100 < 81? No. Since 100 > 81, this triangle is acute. So, Option D is not an obtuse triangle.

Option E: 3, 5, 6

Finally, let's check the Triangle Inequality Theorem. The longest side is 6.

  • Is 3 + 5 > 6? Yes, 8 > 6.
  • Is 3 + 6 > 5? Yes, 9 > 5.
  • Is 5 + 6 > 3? Yes, 11 > 3.

These lengths can form a triangle. Now, let's apply the Obtuse Angle Test. Our longest side c is 6. So, a = 3 and b = 5.

  • Calculate a² + b²: 3² + 5² = 9 + 25 = 34.
  • Calculate c²: 6² = 36.
  • Compare: Is a² + b² < c²? Is 34 < 36? YES! Since 34 < 36, this triangle is obtuse. Another winner, folks! Option E is also an obtuse triangle! The angle opposite the side of length 6 is the one that's bigger than 90 degrees.

The Verdict!

So, after all that number crunching and testing, which sets of three numbers represent the sides of an obtuse triangle? We found two!

  • Option C: 2, 2, 3
  • Option E: 3, 5, 6

These are the sets where the lengths satisfy the Triangle Inequality Theorem and the square of the longest side is greater than the sum of the squares of the other two sides (a² + b² < c²). Remember these rules, guys, they're your golden ticket to identifying obtuse triangles (and acute and right ones too!). Keep practicing, and you'll be a triangle expert in no time!