Hypotenuse Length: A Right Triangle Challenge

by Kenji Nakamura 46 views

Hey guys! Let's dive into a super interesting geometry problem that involves right triangles. We're going to break down the question, explore the concepts, and find the correct answer together. Trust me, by the end of this article, you'll be a pro at tackling these types of problems. So, grab your thinking caps, and let's get started!

The Challenge: Unveiling the Hypotenuse

Our main task is to find the length of the hypotenuse in a right triangle. We know a couple of key things: the shortest side measures 33{sqrt{3]}$ inches and one of the angles is 60∘60^{\circ}. Seems a bit tricky at first, right? But don't worry, we'll break it down step by step. Understanding the relationships between sides and angles in right triangles is crucial here. Think about those special triangles – they're going to be our best friends in solving this problem. We need to use the information given to us strategically. Identifying the type of right triangle we're dealing with is the first step. Is it a 30-60-90 triangle? Or perhaps a 45-45-90 triangle? Knowing this will guide us to the correct ratios for the sides. Let's keep in mind what the hypotenuse actually is – it's the longest side, opposite the right angle. This will be important as we compare our calculated lengths to the possible answers. We'll explore different approaches, like using trigonometric ratios or the properties of special right triangles, to arrive at our final answer. So, let’s keep going and figure out how to decode this triangle!

Cracking the Code: Right Triangle Fundamentals

Before we jump into solving the problem directly, let's refresh some fundamental concepts about right triangles. This will give us a solid base to work from. So, what exactly is a right triangle? It’s a triangle that has one angle measuring exactly 90∘90^{\circ}. This right angle is super important because it gives the triangle some unique properties. Now, let’s talk about the sides of a right triangle. The longest side, which is opposite the right angle, is called the hypotenuse. The other two sides are called legs. The legs are often referred to as the 'opposite' and 'adjacent' sides, depending on which acute angle you are referencing. Understanding the relationship between these sides and angles is key. This is where trigonometric ratios like sine, cosine, and tangent come into play. Remember SOH CAH TOA? It’s a handy mnemonic to remember these ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. Another crucial concept is the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2+b2=c2a^2 + b^2 = c^2). This theorem is a powerful tool for finding the lengths of sides when you know the other two. Special right triangles, like the 30-60-90 and 45-45-90 triangles, are also our friends. They have specific side ratios that make calculations easier. We’ll see how these concepts apply to our problem shortly!

The 30-60-90 Triangle Connection

Okay, guys, let's zoom in on a particularly helpful concept: the 30-60-90 triangle. This is a special right triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees, respectively. What makes it so special? Well, the sides of a 30-60-90 triangle have a very specific ratio. If we let the side opposite the 30-degree angle be 'x', then the side opposite the 60-degree angle is xx{sqrt{3]}$, and the hypotenuse (the side opposite the 90-degree angle) is 2x. This ratio is super important because it allows us to quickly find the lengths of the sides if we know just one side. Now, let's connect this to our problem. We're told that one angle of the right triangle is 60 degrees. This immediately suggests that we might be dealing with a 30-60-90 triangle! We also know that the shortest side measures 33{sqrt{3]}$ inches. The shortest side in a 30-60-90 triangle is the side opposite the 30-degree angle. However, in our case, 33{sqrt{3]}$ inches is opposite the 60-degree angle. This is a crucial detail that helps us identify the value of 'x' in our ratio. By understanding this relationship, we can use the 30-60-90 triangle properties to directly calculate the length of the hypotenuse. Let's dive into the calculations now!

Solving the Puzzle: Finding the Hypotenuse Length

Alright, let's put everything together and calculate the length of the hypotenuse. We know we have a right triangle with a 60-degree angle, and the shortest side (opposite the 60-degree angle) measures 33sqrt{3]}$ inches. From our discussion on 30-60-90 triangles, we know the side opposite the 60-degree angle is xx{sqrt{3]}$. So, we can set up an equation xx{sqrt{3]$ = 3{sqrt{3]}$. Solving for x, we divide both sides by ${sqrt{3]}$, which gives us x = 3. Now, remember the ratio for a 30-60-90 triangle? The hypotenuse is 2x. Since we found x to be 3, the hypotenuse is 2 * 3 = 6 inches. Ta-da! We've found the length of the hypotenuse. Now, let's double-check our answer with the given options. We have A. 6, B. 66{sqrt{3]}$, C. 3, and D. 88{sqrt{2]}$. Our calculated answer, 6 inches, matches option A. So, we're confident that option A is the correct answer. It's always a good idea to review the problem and the steps we took to make sure everything makes sense. We correctly identified the triangle as a 30-60-90 triangle, used the side ratios, and solved for the hypotenuse. Great job, guys!

The Grand Finale: Choosing the Right Answer

So, after carefully dissecting the problem and going through the calculations, we've arrived at our final answer. We were tasked with finding the length of the hypotenuse in a right triangle, given that the shortest side measures 33{sqrt{3]}$ inches and one angle is 60∘60^{\circ}. We identified this as a 30-60-90 triangle, which allowed us to use the specific side ratios for this type of triangle. By setting up the equation xx{sqrt{3]}$ = 3{sqrt{3]}$, we found that x = 3. Then, using the ratio that the hypotenuse is 2x, we calculated the hypotenuse to be 2 * 3 = 6 inches. Now, let's revisit the answer choices one last time:

A. 6 B. 66{sqrt{3]}$ C. 3 D. 88{sqrt{2]}$

Our calculated answer, 6 inches, corresponds perfectly with answer choice A. Therefore, we can confidently say that A. 6 is the correct answer. This whole process demonstrates the power of understanding fundamental geometric principles and how they can be applied to solve problems. Remember, guys, practice makes perfect! Keep working on these types of problems, and you'll become even more confident in your problem-solving skills. Let’s keep the momentum going and explore more exciting math challenges!