Rectangle Sides: Perimeter & Proportionality

Hey math enthusiasts! Let's dive into a classic geometry problem. We're going to figure out how to find the sides of a rectangle when we know its perimeter and the relationship between its length and width. Sounds fun, right? This is a super important concept, and understanding it will help you tackle all sorts of shape problems in the future. So, grab your pencils, and let's get started!

Understanding the Problem: The Foundation of Our Solution

Alright, guys, let's break down what we're dealing with. The problem tells us we have a rectangle. Now, a rectangle, as you probably know, is a four-sided shape with four right angles (90 degrees each). It's got two pairs of equal sides: the longer sides, which we call the length, and the shorter sides, which we call the width. The problem also provides us with a couple of key pieces of information:

• Perimeter: The perimeter of the rectangle is 130 cm. Remember, the perimeter is the total distance around the outside of the shape. Imagine walking around the rectangle; the perimeter is the total distance you'd walk.
• Proportionality: The lengths of the sides are proportional to the numbers 8 and 5. What does proportional mean? It means the ratio between the length and the width is the same as the ratio between 8 and 5. We can write this as length/width = 8/5. This tells us that the length is always a multiple of 8, and the width is always a multiple of 5. These are crucial components, and you'll often see these in geometry challenges. Grasping them is a game-changer.

Now, our mission, should we choose to accept it, is to find the exact length and width of the rectangle. To do this, we'll need to use our knowledge of perimeters and proportions. Keep in mind that understanding these principles is fundamental to tackling other mathematical problems. Geometry concepts are building blocks!

We need to utilize what we already understand about rectangles and apply those principles to discover the lengths of the rectangle's sides. Pay attention to how the concepts connect – that's the secret to mastering these problems!

Setting Up the Equations: Translating Words into Math

Okay, team, let's translate the words of the problem into mathematical equations. This is where we bring the abstract ideas to life. We have two crucial pieces of information that we can turn into equations:

1. Perimeter Equation: We know the perimeter is 130 cm. The formula for the perimeter of a rectangle is: Perimeter = 2 * (length + width). Let's represent the length with 'l' and the width with 'w'. So, our equation becomes: 130 = 2 * (l + w).
2. Proportionality Equation: We know the sides are proportional to 8 and 5. This means we can express the length and width in terms of a common factor, let's call it 'x'. So, we can say: l = 8x and w = 5x. This common factor allows us to relate the length and the width through a single variable.

Now, we have a system of equations. We can use these equations to solve for the unknown values (the length and the width). Note the power of equations! They allow us to distill complex problems into manageable steps. Now, let's substitute the values and uncover the unknowns. Keep in mind: Practice makes perfect. These kinds of problems might seem tricky at first, but with practice, you'll become a pro!

Solving for the Unknowns: Unveiling the Rectangle's Dimensions

Alright, buckle up, because we're about to solve this puzzle! We have our equations, and it's time to find the actual dimensions of the rectangle. Here's how we'll do it:

1. Substitute: We know that l = 8x and w = 5x. Let's substitute these values into our perimeter equation: 130 = 2 * (8x + 5x).
2. Simplify: Now, let's simplify the equation: 130 = 2 * (13x) => 130 = 26x.
3. Isolate x: To find 'x', divide both sides of the equation by 26: x = 130 / 26 => x = 5.

Great job, we have found that x = 5! Now, what does this 'x' mean? It's our scaling factor. It's the number that we multiply the proportions (8 and 5) by to get the actual length and width. So we're really close now to finding the length and the width of the rectangle.

4. Calculate Length and Width: Now that we know x = 5, we can easily find the length and width:
   • l = 8x = 8 * 5 = 40 cm
   • w = 5x = 5 * 5 = 25 cm

Therefore, the length of the rectangle is 40 cm, and the width is 25 cm.

Checking Your Answer: Ensuring Accuracy

Always, always, always check your work! It is a critical component of problem-solving. It's a great habit to prevent silly errors and make sure your answer makes sense. Here's how to check if our answer is correct:

1. Calculate the Perimeter: Use the calculated length and width to calculate the perimeter. Perimeter = 2 * (l + w) = 2 * (40 cm + 25 cm) = 2 * 65 cm = 130 cm.
2. Verify Proportionality: Check if the ratio of the length and width is equal to the ratio of 8 and 5: l/w = 40 cm / 25 cm = 8/5. This confirms that the sides are indeed proportional.

Since our calculated perimeter matches the given perimeter, and the sides maintain the correct proportion, we can be confident that our answer is accurate. That is the importance of checking your answer: it validates our final answer. Congratulations, you’ve successfully solved the problem!

Tips and Tricks: Mastering Rectangle Problems

Want to become a rectangle problem whiz? Here are a few tips and tricks to help you on your geometry journey:

• Draw a Diagram: Always draw a diagram! Sketching a rectangle and labeling the sides can help you visualize the problem and keep track of your information. Visualization is a key element in understanding problems.
• Understand Formulas: Make sure you know the formulas for the perimeter and area of a rectangle. You’ll be using these formulas constantly. Memorizing these formulas is super helpful.
• Practice, Practice, Practice: The more problems you solve, the better you’ll become. Work through different examples to get comfortable with the concepts. Repetition is key to mastering these concepts.
• Look for Keywords: Pay attention to keywords like