45-45-90 Triangle Leg Length: Step-by-Step Solution

Hey there, math enthusiasts! Let's dive into a classic geometry problem involving a special type of right triangle: the 45-45-90 triangle. This triangle, as the name suggests, has angles measuring 45 degrees, 45 degrees, and 90 degrees. It's not just any triangle; it's a right isosceles triangle, meaning it has a right angle and two equal sides. Understanding the relationships within this triangle can unlock many geometrical puzzles. In this article, we'll break down how to find the length of a leg when we know the hypotenuse. So, if you're scratching your head over this, don't worry! We'll make it crystal clear. Let's get started!

The 45-45-90 Triangle: A Quick Review

Before we tackle the problem, let's refresh our understanding of the 45-45-90 triangle. This special right triangle pops up frequently in geometry and trigonometry, so knowing its properties is super helpful. The most important thing to remember is the ratio of its sides. If the two legs (the sides opposite the 45-degree angles) have a length of 'x', then the hypotenuse (the side opposite the 90-degree angle) has a length of 'x√2'. This ratio is the key to solving many 45-45-90 triangle problems. Think of it as your secret weapon! Knowing this ratio allows you to quickly find missing side lengths without having to resort to more complex methods like the Pythagorean theorem every time, although that will certainly work as well. This shortcut can save you time and effort on tests and homework, making it a valuable tool in your mathematical arsenal. So, keep that ratio in mind as we move forward and apply it to our specific problem. It's the foundation for understanding how these triangles work and for solving for unknown sides.

Problem Breakdown: Hypotenuse to Leg

Now, let's dissect the problem at hand. We're given a 45-45-90 triangle where the hypotenuse measures 22√2 units. Our mission, should we choose to accept it, is to find the length of one of the legs. Remember, in a 45-45-90 triangle, both legs are congruent, meaning they have the same length. So, if we find the length of one leg, we've found the length of both! This is a crucial piece of information because it simplifies our task considerably. We don't have to worry about calculating two different lengths; we just need to find one. This symmetry is a characteristic feature of the 45-45-90 triangle, stemming from its isosceles nature. Recognizing this property is key to efficiently solving problems involving these triangles. It allows us to focus our efforts on finding a single value, knowing that it applies to both legs. With this understanding, we can approach the problem with a clear strategy, using the unique relationships within the triangle to our advantage.

Applying the Ratio: The Solution

Here's where the magic happens! We know the ratio of sides in a 45-45-90 triangle: leg : leg : hypotenuse = x : x : x√2. We also know that the hypotenuse in our specific triangle is 22√2 units. So, we can set up an equation: x√2 = 22√2. To find 'x' (the length of a leg), we simply need to isolate 'x' on one side of the equation. We can do this by dividing both sides of the equation by √2. This is a standard algebraic technique that allows us to undo the multiplication and solve for our unknown variable. When we perform this division, the √2 on the left side cancels out, leaving us with 'x'. On the right side, the √2 in the numerator and denominator also cancel out, simplifying the expression. This cancellation is a direct result of the properties of square roots and division, making the simplification process straightforward and elegant. The result gives us the value of 'x', which directly corresponds to the length of one leg of the triangle.

Step-by-Step Calculation:

1. Start with the equation: x√2 = 22√2
2. Divide both sides by √2: (x√2) / √2 = (22√2) / √2
3. Simplify: x = 22

Therefore, the length of one leg of the triangle is 22 units. Option C is the correct answer!

Why Other Options Are Incorrect

It's important not only to find the correct answer but also to understand why the other options are incorrect. This helps solidify your understanding of the concept and prevents you from making similar mistakes in the future. Let's take a look at why options A, B, and D are not the right answers:

• Option A: 11 units - This is incorrect because it doesn't account for the √2 relationship between the leg and the hypotenuse in a 45-45-90 triangle. If the leg were 11 units, the hypotenuse would be 11√2 units, not 22√2 units.
• Option B: 11√2 units - This is also incorrect. It seems like this option might be confusing the relationship or perhaps halving the hypotenuse incorrectly. Remember, the leg is shorter than the hypotenuse in a right triangle.
• Option D: 22√2 units - This is the length of the hypotenuse, not the leg. Choosing this option would mean confusing the given information with the desired result. It's a classic mistake to watch out for!

Understanding these incorrect options helps you grasp the specific relationships within the 45-45-90 triangle and avoid common pitfalls. It reinforces the importance of using the correct ratios and applying them accurately to the given problem. By analyzing why certain answers are wrong, you deepen your understanding and improve your problem-solving skills.

Key Takeaways and Tips for Success

Let's recap the key takeaways from this problem and discuss some tips for tackling similar questions in the future. The most important concept to remember is the side ratio of a 45-45-90 triangle: x : x : x√2. This ratio is your best friend when solving these types of problems. Whenever you encounter a 45-45-90 triangle, immediately recall this ratio and how it relates the legs and the hypotenuse. It's the foundation for solving for any missing side length.

Here are some additional tips for success:

• Visualize the triangle: Draw a diagram of the triangle and label the given information. This visual representation can often make the problem clearer and easier to understand.
• Identify the known and unknown: Clearly identify what information you're given (e.g., the hypotenuse) and what you need to find (e.g., the leg length). This helps you focus your efforts and choose the right approach.
• Set up the equation correctly: Ensure you set up the equation using the correct ratio and the given values. Double-check your work to avoid simple errors.
• Simplify carefully: When solving for the unknown, simplify the equation carefully, paying attention to the rules of algebra and square roots. Avoid rushing this step, as errors in simplification can lead to incorrect answers.
• Check your answer: Once you've found a solution, check if it makes sense in the context of the problem. Is the leg length shorter than the hypotenuse? Does the answer fit the ratio? This final check can help you catch any mistakes and ensure your answer is accurate.

By mastering the 45-45-90 triangle ratio and following these tips, you'll be well-equipped to solve a wide range of geometry problems with confidence and accuracy. Remember, practice makes perfect, so keep working on these types of problems to solidify your understanding and improve your skills.

Conclusion: Mastering the 45-45-90 Triangle

Congratulations! You've successfully navigated the world of 45-45-90 triangles and learned how to find the leg length when given the hypotenuse. The key takeaway is the special ratio that governs these triangles: x : x : x√2. Remember this ratio, practice applying it, and you'll be able to solve these problems with ease. Geometry might seem daunting at times, but breaking down complex problems into simpler steps, like we did here, can make it much more manageable. Keep practicing, keep exploring, and you'll continue to grow your mathematical skills. And remember, every problem solved is a step forward in your learning journey!

For further exploration and practice with right triangles, you can visit resources like Khan Academy's Trigonometry section.