45-45-90 Triangle: Find The Leg Length Easily

Hey guys! Let's dive into a classic geometry problem involving a special type of triangle: the $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle. These triangles pop up frequently in math problems, and understanding their properties can make solving them a breeze. In this article, we'll break down the question: "The hypotenuse of a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle measures 4 cm. What is the length of one leg of the triangle?" We'll explore the concepts, the math, and how to arrive at the correct answer. So, buckle up, and let's get started!

Understanding $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ Triangles

First things first, what exactly is a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle? It's a right triangle where two of the angles are $45^{\circ}$ and the third angle, of course, is $90^{\circ}$. The magic of this triangle lies in its side ratios. Because the two angles are equal, the sides opposite them (the legs) are also equal in length. This makes it an isosceles right triangle. The ratio of the sides in a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle is a fundamental concept to grasp. Imagine the two legs of the triangle each having a length of 'x'. The hypotenuse (the side opposite the $90^{\circ}$ angle) then has a length of $x\sqrt{2}$. This $1:1:\sqrt{2}$ ratio is the key to solving many problems involving these triangles. Knowing this ratio allows us to quickly find the missing side lengths if we know just one side. For example, if we know the length of a leg, we can find the hypotenuse by multiplying that length by $\sqrt{2}$. Conversely, if we know the hypotenuse, we can find the length of a leg by dividing the hypotenuse length by $\sqrt{2}$. This simple relationship is a powerful tool in geometry and trigonometry. Understanding this basic principle will not only help you solve this particular problem but will also be invaluable for tackling more complex problems involving right triangles and trigonometric functions.

Breaking Down the Problem

Let's revisit the problem: The hypotenuse of a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle measures 4 cm. We need to find the length of one leg. We've already established the crucial relationship in a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle: the ratio of the sides is $1:1:\sqrt{2}$. This means if we let the length of each leg be 'x', then the hypotenuse is $x\sqrt{2}$. The problem tells us the hypotenuse is 4 cm. So, we can set up an equation: $x\sqrt{2} = 4$. Now, our goal is to isolate 'x' to find the length of a leg. To do this, we'll divide both sides of the equation by $\sqrt{2}$: $x = \frac{4}{\sqrt{2}}$. At this stage, we've found the value of x, which represents the length of one leg. However, the answer might not look quite like the options provided, as it often requires a bit of simplification. The next step involves rationalizing the denominator, which is a common practice in mathematics to make expressions cleaner and easier to work with. We'll tackle that in the following section.

Solving for the Leg Length: Rationalizing the Denominator

We've arrived at the equation $x = \frac{4}{\sqrt{2}}$. Now, we need to rationalize the denominator. This means getting rid of the square root in the bottom of the fraction. The trick is to multiply both the numerator (top) and the denominator (bottom) by $\sqrt{2}$. This doesn't change the value of the fraction because we're essentially multiplying by 1 ($\frac{\sqrt{2}}{\sqrt{2}}$ is equal to 1). So, let's do it: $x = \frac{4}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}$. Now we're getting somewhere! We have a whole number in the denominator. We can simplify this fraction further. Notice that both the numerator and the denominator are divisible by 2. Dividing both by 2, we get: $x = \frac{2\sqrt{2}}{1} = 2\sqrt{2}$. Therefore, the length of one leg of the triangle is $2\sqrt{2}$ cm. We've successfully used the properties of $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangles and a little bit of algebra to find the answer. Rationalizing the denominator is a crucial skill in simplifying radical expressions and is frequently used in solving geometry and trigonometry problems. Understanding this process ensures that your answers are in their simplest form and easily comparable to answer choices provided.

The Answer and Why It Matters

So, the length of one leg of the triangle is $2\sqrt{2}$ cm. Looking back at the options, the correct answer is B. $2 \sqrt{2}$ cm. We found this by using the special properties of the $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle and applying a bit of algebraic manipulation. But why is this important? Well, understanding special right triangles and their side ratios is crucial in various areas of math and science. These triangles appear frequently in geometry problems, trigonometry, and even in fields like physics and engineering. Being able to quickly recognize and work with these triangles can save you a lot of time and effort. Moreover, the process of solving this problem highlights important mathematical skills such as setting up equations, simplifying radicals, and rationalizing denominators. These skills are fundamental building blocks for more advanced mathematical concepts. So, mastering these concepts not only helps you ace geometry problems but also prepares you for future mathematical challenges. Whether you're calculating distances, designing structures, or analyzing forces, the principles we've discussed here have wide-ranging applications.

Key Takeaways and Tips

Let's recap the key takeaways from this problem: $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangles have a special side ratio of $1:1:\sqrt{2}$. This is your golden ticket to solving problems involving these triangles. If you know one side, you can find the others! Remember to rationalize the denominator when you have a square root in the denominator. This is a standard practice for simplifying expressions. Practice, practice, practice! The more you work with these types of problems, the more comfortable you'll become with recognizing patterns and applying the correct techniques. Here are a few extra tips to help you master $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangles: Visualize the triangle. Draw a diagram if one isn't provided. This helps you see the relationships between the sides and angles. Memorize the side ratio. It's a fundamental concept that will save you time. Look for opportunities to apply the Pythagorean Theorem as a backup. While the side ratio is the most efficient method for $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangles, the Pythagorean Theorem ($a^2 + b^2 = c^2$) always works for right triangles. By mastering these key concepts and practicing regularly, you'll be well-equipped to tackle any $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle problem that comes your way. Keep up the great work, guys!

In conclusion, we've successfully navigated this geometry problem by understanding the properties of $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangles, setting up an equation, and simplifying the result. Remember the $1:1:\sqrt{2}$ ratio, and you'll be a pro at these in no time! Keep practicing, and you'll conquer any geometry challenge.