Right Triangle Sides: Perimeter 80 Calculation Explained
Hey guys! Ever wondered how to figure out the sides of a right-angled triangle when you know its perimeter? It's a common question in mathematics, and in this article, we're diving deep into solving exactly that! We'll break down the problem step-by-step, making it super easy to understand. Let’s explore how to identify the possible sides of a right-angled triangle when the perimeter is given as 80. This is a classic problem that combines the concepts of perimeter and the Pythagorean theorem, so buckle up and let's get started!
Understanding the Basics
Before we jump into solving the problem, let's quickly recap some essential concepts. This will ensure we're all on the same page and ready to tackle the challenge. Understanding these basics is crucial for grasping the solution and applying it to similar problems. Remember, a strong foundation is key to mastering any mathematical concept!
What is a Right-Angled Triangle?
A right-angled triangle, as the name suggests, is a triangle with one angle measuring exactly 90 degrees. This 90-degree angle is often called a right angle. The side opposite the right angle is the longest side and is known as the hypotenuse. The other two sides are called legs or cathetus. Visualizing a right-angled triangle is the first step in understanding its properties and how they relate to its perimeter and side lengths.
The Pythagorean Theorem
The Pythagorean Theorem is a fundamental concept in geometry that applies specifically to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it’s expressed as:
a² + b² = c²
Where:
* *a* and *b* are the lengths of the two shorter sides (legs).
* *c* is the length of the hypotenuse.
This theorem is essential for solving problems involving right-angled triangles, especially when dealing with side lengths. We'll be using this extensively to check if a given set of sides can form a right-angled triangle.
What is the Perimeter of a Triangle?
The perimeter of any polygon, including a triangle, is the total length of all its sides. For a triangle, this simply means adding the lengths of the three sides together. If we denote the sides of a triangle as a, b, and c, then the perimeter P is given by:
P = a + b + c
In our problem, we know the perimeter is 80, so we have a + b + c = 80. This equation will be crucial in verifying the possible sets of sides.
How These Concepts Connect
The magic happens when we combine the concept of the perimeter with the Pythagorean Theorem. We know the total length of the sides (perimeter), and we have a relationship between the sides of a right-angled triangle (Pythagorean Theorem). By using both, we can determine if a set of given sides can form a right-angled triangle with the specified perimeter. It's like having two pieces of a puzzle that fit together perfectly! Understanding this connection is key to solving our problem efficiently.
Analyzing the Given Options
Now that we have a solid understanding of the basic concepts, let's dive into the specific problem. We're given four sets of side lengths and need to determine which one can form a right-angled triangle with a perimeter of 80. Remember, we need to check two things for each set of sides:
- Does the sum of the sides equal 80 (the given perimeter)?
- Do the sides satisfy the Pythagorean Theorem (a² + b² = c²)?
Let's go through each option step-by-step. This methodical approach will help us avoid mistakes and ensure we find the correct answer. Get ready to put on your detective hats, guys!
Option 1: 18, 25, 37
First, let's check if the sum of the sides equals 80:
18 + 25 + 37 = 80
Great! The perimeter condition is satisfied. Now, let’s see if these sides can form a right-angled triangle using the Pythagorean Theorem. We need to identify the longest side, which is 37, and treat it as the hypotenuse (c). The other two sides, 18 and 25, will be a and b.
Applying the theorem:
18² + 25² = 37²
Let's calculate the squares:
324 + 625 = 1369
949 ≠1369
The equation does not hold true. Therefore, the sides 18, 25, and 37 cannot form a right-angled triangle. So, this option is out! We're one step closer to finding the right answer.
Option 2: 15, 31, 34
Let's start by checking the perimeter:
15 + 31 + 34 = 80
Again, the perimeter condition is met. Now, let's check the Pythagorean Theorem. The longest side is 34, so it will be our c (hypotenuse), and 15 and 31 will be a and b.
Applying the theorem:
15² + 31² = 34²
Calculate the squares:
225 + 961 = 1156
1186 ≠1156
This equation also doesn't hold true. The sides 15, 31, and 34 cannot form a right-angled triangle. Don't worry, guys, we're narrowing down our options!
Option 3: 16, 30, 34
Let's check the perimeter first:
16 + 30 + 34 = 80
Perfect! The perimeter matches. Now, let's apply the Pythagorean Theorem. The longest side is 34, so it’s our c, and 16 and 30 are a and b.
Using the theorem:
16² + 30² = 34²
Calculate the squares:
256 + 900 = 1156
1156 = 1156
This equation holds true! The sides 16, 30, and 34 can form a right-angled triangle. It looks like we've found our answer, but let's check the last option just to be sure. It's always good to double-check, right?
Option 4: 11, 34, 35
Check the perimeter:
11 + 34 + 35 = 80
The perimeter condition is satisfied. Now, let's check the Pythagorean Theorem. The longest side is 35, so it's our c, and 11 and 34 are a and b.
Applying the theorem:
11² + 34² = 35²
Calculate the squares:
121 + 1156 = 1225
1277 ≠1225
This equation doesn't hold true. The sides 11, 34, and 35 cannot form a right-angled triangle. So, we've confirmed that our answer is indeed correct!
Conclusion
After carefully analyzing each option, we've found that the sides 16, 30, and 34 are the ones that can form a right-angled triangle with a perimeter of 80. This was determined by verifying that the sum of the sides equals 80 and that they satisfy the Pythagorean Theorem. See, guys? Math can be pretty cool when you break it down step by step!
Key Takeaways
- Always start by understanding the basic concepts: In this case, right-angled triangles, the Pythagorean Theorem, and perimeter.
- Break the problem into smaller, manageable steps: Check the perimeter first, then apply the Pythagorean Theorem.
- Be methodical: Analyze each option systematically to avoid errors.
- Double-check your answer: It’s always a good idea to verify your solution.
Final Thoughts
Solving problems like these not only enhances your understanding of mathematical concepts but also improves your problem-solving skills in general. Keep practicing, guys, and you'll become math whizzes in no time! Remember, every problem is a chance to learn something new and sharpen your mind. So, keep those brains buzzing and keep exploring the fascinating world of mathematics!
