Right Triangle Sunshade: Find Leg Lengths

Dec 8, 2025 by Alex Johnson 42 views

Have you ever looked at a stylish sunshade and wondered about the math behind its design? Often, these shades are shaped like right isosceles triangles, providing both form and function. If you've encountered a problem where the equation $rac{1}{2} x^2 = 64$ represents one such shade, shielding a generous 64 square feet of area, you might be asking yourself: which system can be used to find the lengths of the legs of this sun shade? This isn't just about abstract numbers; it's about understanding how geometric shapes and algebraic equations work together in practical applications. We'll delve into how to solve this, breaking down the concepts so you can see the relationship between area, equations, and the dimensions of a right isosceles triangle. Let's get started on unraveling this geometrical puzzle and discover the elegant simplicity of finding those leg lengths!

Understanding the Geometry of Your Sunshade

Let's talk about the shape of your sunshade. When we say it’s a right isosceles triangle, we're describing a very specific kind of triangle. A right triangle, as you probably know, has one angle that measures exactly 90 degrees – that’s the ‘right’ part. An isosceles triangle is one that has two sides of equal length. So, a right isosceles triangle is a triangle that has both a right angle and two sides of equal length. In this special type of triangle, the two equal sides are always the ones that form the right angle. These two equal sides are often referred to as the 'legs' of the right triangle. The third side, the longest one opposite the right angle, is called the hypotenuse. For our sunshade problem, the legs are the key pieces of information we need to find, as they define the extent of the shade.

Now, let's connect this to the area. The area of any triangle is calculated by the formula: Area = $rac{1}{2} imes ext{base} imes ext{height}$ . In a right isosceles triangle, since the two legs are equal in length and form the right angle, we can use one leg as the base and the other leg as the height. If we let the length of each leg be represented by the variable $x$ , then the base is $x$ and the height is also $x$ . Plugging this into the area formula, we get: Area = $rac{1}{2} imes x imes x$ , which simplifies to Area = $rac{1}{2} x^2$ . This is precisely the form of the equation given in our problem: $rac{1}{2} x^2 = 64$ . This equation directly relates the area of the sunshade (64 square feet) to the length of its equal legs ( $x$ ). So, when you see $rac{1}{2} x^2 = 64$ , you're looking at the mathematical representation of the sunshade's area, where $x$ is the length of each of the two equal sides that form the right angle. The problem is essentially asking us to solve this equation for $x$ to find those crucial leg lengths.

Decoding the Equation: $rac{1}{2} x^2 = 64$

Let's focus on the given equation: $rac{1}{2} x^2 = 64$ . As we've established, this equation is specifically designed to help us find the lengths of the legs of our right isosceles triangle sunshade. The '64' in the equation represents the total area the shade covers, given as 64 square feet. The $rac{1}{2} x^2$ part is the formula for the area of a right isosceles triangle where $x$ is the length of each of the two equal legs. Our goal now is to isolate $x$ – to get it all by itself on one side of the equation – so we can determine its value. This process of isolating a variable is fundamental to solving algebraic equations and is a key skill in mathematics.

To start solving for $x$ , we first need to get rid of the $rac{1}{2}$ that's multiplying the $x^2$ . The opposite operation of multiplying by $rac{1}{2}$ is multiplying by 2. So, we'll multiply both sides of the equation by 2. This gives us: $2 imes ( rac{1}{2} x^2) = 2 imes 64$ . Simplifying this, the 2 and $rac{1}{2}$ on the left side cancel each other out, leaving us with $x^2$ . On the right side, $2 imes 64$ equals 128. So, our equation now becomes $x^2 = 128$ . This tells us that the square of the length of one leg, when multiplied by itself, equals 128.

Now, we need to find the value of $x$ . Since $x^2 = 128$ , $x$ must be the number that, when multiplied by itself, gives us 128. This is the definition of a square root. Therefore, we need to find the square root of 128. Mathematically, we write this as $x = e oxed{ ext{sqrt}}(128)$ . When we calculate the square root of 128, we find that it's approximately 11.31. So, the length of each leg of the sunshade is approximately 11.31 feet. This process of algebraic manipulation is how we decode the equation and arrive at the dimensions of our sunshade, showcasing the power of mathematics in solving real-world problems.

Systems for Finding Leg Lengths: A Deeper Dive

When tackling a problem like finding the lengths of the legs of our right isosceles triangle sunshade, represented by the equation $rac{1}{2} x^2 = 64$ , we are essentially looking for a system that allows us to solve for the unknown variable, $x$ . The question asks which system can be used, and in this context, a 'system' refers to the set of mathematical principles and operations that guide us toward the solution. The most direct and fundamental system here is the algebraic system of solving equations. This system involves a series of logical steps to isolate the variable and determine its value.

Our starting point is the equation $rac{1}{2} x^2 = 64$ . The primary goal within this algebraic system is to determine the value of $x$ . To achieve this, we employ inverse operations. First, to undo the multiplication by $rac{1}{2}$ , we multiply both sides of the equation by 2. This maintains the equality of the equation. So, we have $2 imes ( rac{1}{2} x^2) = 2 imes 64$ , which simplifies to $x^2 = 128$ . This step effectively isolates the $x^2$ term.

The next step involves undoing the squaring of $x$ . The inverse operation of squaring a number is taking its square root. Therefore, we take the square root of both sides of the equation: $ e oxed ext{sqrt}}(x^2) = e oxed{ ext{sqrt}}(128)$. This gives us $x = e oxed{ ext{sqrt}}(128)$ . It's important to note that when we solve $x^2 = k$ , there are technically two solutions a positive and a negative one. However, in the context of a physical length, such as the leg of a sunshade, the length must be a positive value. Therefore, we only consider the positive square root. The value of $ e oxed{ ext{sqrt}(128)$ can be simplified. We know that $128 = 64 imes 2$ . So, $ e oxed{ ext{sqrt}}(128) = e oxed{ ext{sqrt}}(64 imes 2) = e oxed{ ext{sqrt}}(64) imes e oxed{ ext{sqrt}}(2) = 8 e oxed{ ext{sqrt}}(2)$.

Thus, the system used is the fundamental principles of algebra, specifically the manipulation of equations using inverse operations (multiplication and square root). This system allows us to transform the given area equation into a form where the length of the leg, $x$ , can be directly calculated. The solution $x = 8 e oxed{ ext{sqrt}}(2)$ feet provides the exact length. If an approximate decimal value is needed, we can calculate $8 imes 1.414...$ , which is approximately 11.31 feet. This entire process demonstrates a coherent system of mathematical reasoning applied to a geometric problem.

Practical Application and Conclusion

We've journeyed through understanding the shape of the sunshade, decoding its associated equation, and identifying the mathematical system used to solve for the lengths of its legs. The core of this problem lies in the relationship between the area of a right isosceles triangle and its dimensions. The equation $rac{1}{2} x^2 = 64$ elegantly encapsulates this relationship, where $x$ represents the length of each equal leg. By applying the fundamental principles of algebra – specifically, the use of inverse operations like multiplication and taking the square root – we successfully isolated $x$ and found its value.

The system employed is the algebraic method of equation solving. This involves a sequence of steps: first, multiplying both sides of the equation by 2 to eliminate the fraction, leading to $x^2 = 128$ . Second, taking the square root of both sides to solve for $x$ , resulting in $x = e oxed{ ext{sqrt}}(128)$ . As a length must be positive, we consider only the positive root. The exact answer is $x = 8 e oxed{ ext{sqrt}}(2)$ feet. This demonstrates a powerful application of mathematics in determining the physical dimensions of an object based on its area and shape.

This problem highlights how abstract mathematical concepts have tangible applications in our everyday lives, from the design of home furnishings like sunshades to more complex engineering feats. Understanding these principles not only helps in solving specific problems but also builds a foundation for tackling more intricate challenges in science, technology, engineering, and mathematics (STEM) fields. The ability to translate a real-world scenario into a mathematical model and then solve that model is a crucial skill.

If you're interested in learning more about the geometry of shapes or the power of algebraic problem-solving, resources like Khan Academy offer a wealth of information and practice. They provide excellent tutorials and exercises that can deepen your understanding of these fundamental mathematical concepts. Exploring these platforms can further illuminate the practical beauty of mathematics.