Finding Garden Width Using Pythagorean Theorem

Dec 5, 2025 by Alex Johnson 47 views

Are you ready to dig into some geometry and gardening? A landscaper is planning a diagonal path through a rectangular garden, and we're going to use the Pythagorean theorem to figure out some key dimensions. It's a fun blend of math and real-world application, perfect for anyone who loves a good puzzle or enjoys sprucing up their outdoor space. So, grab your calculator (or your phone!) and let's get started. We'll be using concepts like the diagonal of a rectangle, the relationship between the width and length, and of course, the ever-useful Pythagorean theorem. This is going to be a fun journey where we combine mathematics with the practical aspects of landscaping and gardening. Get ready to transform abstract concepts into tangible results. Whether you're a math enthusiast or just curious about how geometry applies to everyday life, this exploration will provide valuable insights and practical skills. Let's delve in!

Understanding the Problem: The Garden's Design

Our landscaper is designing a path across a rectangular garden. Imagine a beautiful garden, perhaps your own, or one you dream of having. The path will run diagonally from one corner to the opposite corner. This diagonal path is 12 feet long. We also know some details about the garden itself: the width of the garden is represented by the variable x feet, and the length of the garden is 4 feet more than the width. This is where the beauty of algebra comes into play – we can translate this information into mathematical expressions. We’re working with a right triangle within the rectangle. The diagonal path forms the hypotenuse, and the width and length of the garden are the other two sides. The Pythagorean theorem, which we will use to solve this problem, tells us how these sides relate to each other.

So, what are we trying to find? The landscaper wants to determine the width of the garden, represented by x. To find this, we’ll use the Pythagorean theorem: a² + b² = c². In this equation, a and b are the lengths of the two shorter sides of the right triangle (the width and length of the garden), and c is the length of the longest side (the diagonal path). We will need to set up an equation, using the given information to calculate the value of x. The equation will involve squaring the width and length, adding those squares together, and setting the result equal to the square of the diagonal's length. This will allow us to find the specific value of x. This is the crux of our problem: how to transform words and dimensions into an equation that we can solve. The end goal is to use the math to discover an unknown measurement – the width of the garden.

Setting Up the Equation

Let's put our thinking caps on and set up the equation that will help us find the width of the garden. Remember, the Pythagorean theorem states: a² + b² = c². In our garden scenario:

The width of the garden is x (this will be our a side).
The length of the garden is x + 4 (this will be our b side, as it is 4 feet longer than the width).
The diagonal path, or hypotenuse, is 12 feet (this is our c side).

Plugging these values into the Pythagorean theorem, we get: x² + (x + 4)² = 12². Now, we have a quadratic equation. Solving this equation will require expanding, simplifying, and then isolating the variable x. The process will allow us to apply algebra to solve a practical geometry problem. We will use the formula above and apply the given data. This step-by-step approach simplifies the complex concept into a digestible solution. Let's begin the exciting journey of using math to understand the geometry of a garden. This is a very interesting real-world application of mathematics. Let’s proceed to the next step and continue the process.

Solving for the Width: Step-by-Step

Now, let's solve the equation x² + (x + 4)² = 12² step by step. This is where we put our algebra skills to work. Breaking down the process into smaller, manageable steps will make it easier to follow and understand.

Expand the Equation: First, we need to expand the expression (x + 4)². Remember, this means (x + 4) * (x + 4). Using the FOIL method (First, Outer, Inner, Last), we get: x² + 4x + 4x + 16. So, (x + 4)² becomes x² + 8x + 16.
Rewrite the equation: Now, substitute this back into our original equation: x² + (x² + 8x + 16) = 12². We also know that 12² is 144, so the equation is: x² + x² + 8x + 16 = 144.
Simplify and Combine Like Terms: Combine the x² terms and simplify the equation: 2x² + 8x + 16 = 144. Then, subtract 144 from both sides to set the equation to zero: 2x² + 8x - 128 = 0.
Simplify the Equation: To make the equation easier to work with, we can divide the entire equation by 2: x² + 4x - 64 = 0. This simplifies the coefficients and will make the next step easier.
Solve the Quadratic Equation: Now, we have a quadratic equation. We can solve it using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. In our equation, a = 1, b = 4, and c = -64. Plugging these values into the quadratic formula, we get: x = (-4 ± √(4² - 4 * 1 * -64)) / 2 * 1. This yields the solutions for x.

After carrying out these steps, the next part involves calculating the solution of the quadratic equation. Then, we can evaluate each step and ensure that it is accurate, so the final result is correct. This is the application of algebra, and a solid understanding can lead to great results. The ability to solve a quadratic equation will help us find the dimensions of the garden.

Finding the Solution

Let’s finish up and crunch the numbers. Here’s what we get after using the quadratic formula to solve for x: x = (-4 ± √(16 + 256)) / 2 which simplifies to x = (-4 ± √272) / 2. Now, calculate the square root of 272 (approximately 16.49). Therefore, we have two possible solutions:

x = (-4 + 16.49) / 2 ≈ 6.25 feet
x = (-4 - 16.49) / 2 ≈ -10.25 feet

Since the width of a garden can't be negative, we disregard the negative solution. The only plausible answer for x is about 6.25 feet. This calculation gives us the width of the garden that the landscaper wants to build. We are ready to make a conclusion and verify this answer. This real-world situation can be solved using mathematical principles. The final answer tells us the width of the garden. That is the ultimate goal!

Conclusion: The Width of the Garden

After all our calculations, the width of the garden is approximately 6.25 feet. Since the length is 4 feet more than the width, the length of the garden would be approximately 10.25 feet. This provides us with all of the dimensions needed to build the garden. And now, the diagonal path? It will measure exactly 12 feet, as planned. The ability to use the Pythagorean theorem gives us the tools to analyze and build structures. This makes the creation of a garden a rewarding process. The mathematical concepts are used here to create a beautiful garden. This is a very satisfying result.

This exercise clearly demonstrates how the Pythagorean theorem isn't just a classroom concept; it's a practical tool. It is useful for landscapers and anyone involved in construction or design. By understanding and applying this theorem, the landscaper can accurately plan the path and ensure it fits perfectly within the garden's dimensions. It's about taking the theoretical and making it real, turning abstract concepts into a tangible outcome. With these skills, you can approach similar problems with confidence. It's a great example of how math skills can be applied to real-world problems. Whether you're planning a garden, building a house, or simply curious about how things work, mathematics provides a framework for understanding and solving problems. This helps make our world a much better place!

For further reading and more examples of how the Pythagorean theorem is used in practical applications, check out these trusted resources:

Khan Academy - Pythagorean Theorem: This website provides excellent educational videos and practice exercises on the Pythagorean Theorem. This is a very great place for learning.
Math is Fun - Pythagorean Theorem: Another wonderful source that offers clear explanations and interactive examples related to the Pythagorean theorem and related concepts. This is also a good website for learning.