Right Triangle Check: Side Lengths & Pythagorean Theorem
Hey guys! Let's dive into a fundamental concept in geometry: right triangles. Today, we're tackling the question of how to determine if a triangle is a right triangle just by knowing the lengths of its sides. This involves using a super important theorem called the Pythagorean Theorem. We'll walk through it step-by-step, making sure you grasp the core idea and can apply it yourself. So, buckle up, and let's get started!
Understanding the Pythagorean Theorem
At the heart of determining whether a triangle is a right triangle lies the Pythagorean Theorem. This theorem is a cornerstone of geometry and provides a relationship between the sides of a right triangle. It states that in a right triangle, 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 (legs). Mathematically, this is expressed as:
a² + b² = c²
Where:
aandbare the lengths of the legs of the right triangle.cis the length of the hypotenuse.
Think of it this way: if you square the lengths of the two shorter sides and add them together, the result should be equal to the square of the length of the longest side (the hypotenuse) if the triangle is indeed a right triangle. If this equation holds true, we can confidently say that the triangle is a right triangle. If it doesn't, then the triangle is not a right triangle. This is the fundamental principle we'll be using to solve our problem.
But why is this theorem so important? Well, it allows us to not only identify right triangles but also to calculate missing side lengths if we know the other two sides. This is incredibly useful in various fields, from construction and engineering to navigation and even art! Understanding the Pythagorean Theorem opens up a whole new world of geometric problem-solving. So, let's see how we can apply this theorem to the specific triangles given in our problem.
Identifying Right Triangles: The Converse
Now, let's talk about how we actually use the Pythagorean Theorem to check if a triangle is a right triangle. We're essentially using the converse of the theorem. The converse of a statement basically flips the order of the original statement. So, while the Pythagorean Theorem says “if a triangle is a right triangle, then a² + b² = c²”, the converse says “if a² + b² = c², then the triangle is a right triangle.”
This converse is what allows us to work backward. We know the side lengths (a, b, and c), and we want to know if the triangle is a right triangle. So, we plug the side lengths into the equation a² + b² = c² and see if it holds true. Remember, c always represents the longest side (the potential hypotenuse). If the equation is satisfied, bingo! We've got a right triangle. If not, then it's not a right triangle.
This method is super practical because it gives us a concrete way to verify the nature of a triangle without having to measure angles or construct the triangle physically. It's a pure calculation-based approach, which is why it's so valuable in mathematics and related fields. We're going to apply this converse to the two triangles given in the problem. We'll take each set of side lengths, plug them into the equation, and see if the equality holds. This will definitively tell us whether each triangle is a right triangle or not. So, let's move on and tackle the specific examples!
Example (i): a = 6 cm, b = 8 cm, c = 10 cm
Let's put the Pythagorean Theorem to the test with our first triangle, where the sides are given as a = 6 cm, b = 8 cm, and c = 10 cm. Remember, we need to check if a² + b² = c². The first step is to plug in the values we have for a, b, and c into the equation.
So, we have:
6² + 8² = 10²
Now, we need to calculate the squares of each of these numbers. 6 squared (6²) is 6 * 6 = 36. 8 squared (8²) is 8 * 8 = 64. And 10 squared (10²) is 10 * 10 = 100. Let's substitute these values back into our equation:
36 + 64 = 100
The next step is to add the numbers on the left side of the equation. 36 plus 64 equals 100. So, now we have:
100 = 100
Look at that! The left side of the equation is equal to the right side. This means that the Pythagorean Theorem holds true for this triangle. Therefore, we can confidently conclude that the triangle with sides 6 cm, 8 cm, and 10 cm is a right triangle! Awesome! We've successfully identified a right triangle using the theorem. Now, let's move on to the second example and see if it also fits the criteria.
Example (ii): a = 5 cm, b = 8 cm, c = 11 cm
Alright, let's move on to the second triangle. This time, we have sides with lengths a = 5 cm, b = 8 cm, and c = 11 cm. Just like before, we're going to use the Pythagorean Theorem (a² + b² = c²) to check if this triangle is a right triangle. Let's plug in the values for a, b, and c:
5² + 8² = 11²
Now, we need to calculate the squares. 5 squared (5²) is 5 * 5 = 25. 8 squared (8²) is 8 * 8 = 64. And 11 squared (11²) is 11 * 11 = 121. Let's put those values back into the equation:
25 + 64 = 121
Next, we add the numbers on the left side of the equation: 25 plus 64 equals 89. So, our equation now looks like this:
89 = 121
Hmm, this is interesting. 89 is not equal to 121. This means that the Pythagorean Theorem does not hold true for this triangle. Therefore, we can conclude that the triangle with sides 5 cm, 8 cm, and 11 cm is not a right triangle. So, not every triangle will satisfy the Pythagorean Theorem, and that's perfectly okay! It just tells us that it's not a right triangle. Now that we've analyzed both triangles, let's summarize our findings.
Conclusion: Identifying Right Triangles Made Easy
So, guys, we've successfully used the Pythagorean Theorem to determine which of the two given triangles is a right triangle. By applying the converse of the theorem, we were able to take the side lengths and check if they satisfy the equation a² + b² = c². For the first triangle with sides 6 cm, 8 cm, and 10 cm, the equation did hold true, confirming that it is a right triangle. For the second triangle with sides 5 cm, 8 cm, and 11 cm, the equation did not hold true, telling us that it is not a right triangle.
This exercise demonstrates the power and simplicity of the Pythagorean Theorem in identifying right triangles. It's a fundamental tool in geometry and a concept that's worth mastering. Remember, the key is to plug in the side lengths, calculate the squares, and see if the equation balances. If it does, you've got a right triangle! If not, it's a different kind of triangle.
I hope this breakdown has been helpful and clear. Keep practicing with different side lengths, and you'll become a pro at identifying right triangles in no time! Geometry can be fun and it's definitely useful in a lot of ways. Keep exploring and keep learning!