Trapezoidal Frame Perimeter Problem A Step-by-Step Solution

Hey there, math enthusiasts! Let's dive into a fun problem involving shapes and measurements. We've got a real-world scenario here, and by the end of this article, you'll be a pro at solving similar problems. So, grab your thinking caps, and let's get started!

1. Understanding the Problem

So, perimeter problems, like the one we're tackling today, often involve real-life situations. In this case, we're looking at a frame – the kind you might use for a picture or a piece of art. The key to success here is breaking down the problem into smaller, more manageable parts. First, we need to identify the shape we're dealing with. The problem tells us it's a trapezoid. Now, what exactly is a trapezoid? A trapezoid is a four-sided shape, also known as a quadrilateral, with at least one pair of parallel sides. Think of it like a table – the top and bottom are parallel, but the sides might not be. Our frame is in this shape, which means it has four sides. To find the perimeter, we simply need to add up the lengths of all these sides. The problem gives us these lengths: 60 cm, 65 cm, 80 cm, and 95 cm. We've got all the pieces of the puzzle, guys! Understanding what the problem is asking is crucial. In this case, the question is straightforward: What is the perimeter of the frame? But sometimes, problems can be a bit trickier, with extra information or hidden questions. Always make sure you know exactly what you're trying to find before you start crunching numbers. Once we know what we're looking for (the perimeter), and we understand the shape (a trapezoid with given side lengths), we're ready to move on to the next step: planning our solution. This involves thinking about how we can use the information we have to answer the question. For this problem, it's pretty clear: we need to add up the lengths of the sides. But in more complex problems, you might need to use formulas, draw diagrams, or even break the problem down into smaller steps. Now that we've thoroughly understood the problem, let's move on to identifying the key information.

2. Identifying the Given Facts

In any math problem, especially those pesky word problems, identifying the given facts is super important. It's like being a detective and gathering clues! In our trapezoid frame problem, the facts are right there in the description. We know the frame has four sides, and we know the length of each side: 60 cm, 65 cm, 80 cm, and 95 cm. These are our building blocks, the numbers we'll use to find the answer. But identifying the facts isn't just about picking out the numbers. It's also about understanding what those numbers represent. Each measurement corresponds to a side of the trapezoidal frame. Visualizing this can be really helpful. Imagine the frame in your mind, or even sketch it out on paper. Label each side with its length. This way, you're not just looking at numbers; you're seeing the problem in a concrete way. Sometimes, problems will try to trick you with extra information that isn't needed. Learning to sift through the details and focus on what's relevant is a key skill in problem-solving. In our case, we don't need to know what the frame will hold, or what color it is. All we need are the side lengths. Another important aspect of identifying facts is paying attention to the units. In this problem, all the measurements are in centimeters (cm), which makes our job easier. But sometimes, you might have different units (like meters and centimeters) in the same problem. In those cases, you'll need to convert them to the same unit before you can start calculating. For instance, you might need to change meters to centimeters or vice versa. Now that we've identified our given facts – the side lengths of the trapezoid – we're ready to move on to the next crucial step: figuring out what the problem is actually asking us to find. Let's dive in!

3. What the Problem Asks

Okay, let's be clear, pinpointing what the problem is asking is the compass that guides our problem-solving journey. It's like knowing your destination before you start a trip! In our trapezoid frame scenario, the question is pretty straightforward: "What is its perimeter?" But even seemingly simple questions can have layers. To truly understand what we're being asked, we need to break it down. What exactly is "perimeter"? That's the key word here. Perimeter, in the world of geometry, refers to the total distance around the outside of a shape. Think of it like building a fence around a yard – the perimeter is the total length of fencing you'll need. So, in the context of our frame, the perimeter is the total length of the wood (or whatever material it's made of) that forms the frame's outer edges. We're not interested in the area inside the frame, or its height, or anything else – just the distance around the outside. Now, let's relate this back to the shape we're dealing with: a trapezoid. Since a trapezoid has four sides, the perimeter is simply the sum of the lengths of those four sides. This understanding is crucial because it tells us exactly what operation we need to perform: addition. We need to add the lengths of the four sides together to get the perimeter. Sometimes, problems can be a bit more sneaky. They might ask the same question in different ways. For example, instead of saying "What is the perimeter?", they might say "How much material is needed to make the frame?" or "What is the total length of the frame's edges?". These are all different ways of asking for the same thing: the perimeter. Being able to recognize these different phrasings is a valuable skill in problem-solving. Now that we're crystal clear on what the problem is asking – the perimeter, which is the sum of the side lengths – we're ready to put our plan into action and calculate the answer. Let's roll!

4. Solving for the Perimeter

Alright, here comes the fun part – putting all the pieces together and solving for the perimeter! We've identified the shape (trapezoid), the given facts (side lengths: 60 cm, 65 cm, 80 cm, and 95 cm), and what we need to find (perimeter). We also know that the perimeter of any shape is the sum of the lengths of its sides. So, our plan is simple: add the four side lengths of the trapezoid together. This is where our basic math skills come into play. We need to add 60 cm + 65 cm + 80 cm + 95 cm. You can do this in any order you like, as addition is commutative (meaning the order doesn't change the answer). One way to make the addition easier is to group the numbers in a way that's convenient for you. For example, you might add 60 and 80 first, which gives you 140. Then, you could add 65 and 95, which gives you 160. Now, we just need to add 140 and 160, which equals 300. So, the perimeter of the trapezoidal frame is 300 cm. But we're not quite done yet! It's important to always include the units in your answer. In this case, the side lengths were given in centimeters, so our perimeter is also in centimeters. Leaving out the units can sometimes lead to misunderstandings or even incorrect answers in more complex problems. Another important step is to check your answer. Does 300 cm seem like a reasonable perimeter for a frame with sides of those lengths? It does! Our answer makes sense in the context of the problem. Now, let's think about what we've accomplished. We've successfully solved a real-world math problem by breaking it down into smaller, manageable steps. We understood the problem, identified the facts, figured out what we needed to find, made a plan, and executed that plan to arrive at the solution. This is the problem-solving process in action! Now that we've got our answer, let's take a moment to reflect on the problem and see what other questions we can answer.

5. Reflecting on the Solution and Related Questions

Okay, guys, we've nailed the perimeter of the trapezoidal frame – it's 300 cm! But the journey doesn't end with just finding the answer. A true math whiz takes time to reflect on the solution and explore related questions. This deepens our understanding and sharpens our problem-solving skills. First, let's think about what our answer means in the real world. 300 cm is the total length of material needed to make the frame. If we were building this frame, we'd know exactly how much wood (or whatever material we're using) to cut. This is the practical application of math – it helps us solve real-life problems! Now, let's consider some related questions. What if we wanted to put a decorative border around the frame? We'd need to know the perimeter to figure out how much border material to buy. Or, what if we wanted to ship the frame? Knowing the perimeter could help us estimate the size of the box we'd need. We could even think about the area of the trapezoid. While we calculated the perimeter (the distance around the outside), the area tells us the amount of space inside the frame. Calculating the area would involve a different formula, but it's another interesting aspect of the shape. Another valuable exercise is to think about how we could solve the problem using different methods. Could we have used a different order of addition? Could we have broken the trapezoid down into simpler shapes (like rectangles and triangles) to calculate the perimeter? Exploring alternative approaches can give us a deeper understanding of the concepts involved. Finally, let's consider how we could apply this problem-solving process to other situations. We broke down this problem into smaller steps: understanding the problem, identifying facts, figuring out what to find, making a plan, executing the plan, and reflecting on the solution. This is a powerful framework that can be used to tackle all sorts of problems, both in math and in everyday life. By practicing these steps, we become more confident and effective problem-solvers. So, congratulations on mastering the perimeter of the trapezoidal frame! You've not only solved a math problem, but you've also honed your critical thinking skills. Keep exploring, keep questioning, and keep having fun with math!

a. Who bought a frame?

My brother bought the frame.

b. What is the shape of the frame?

The shape of the frame is trapezoidal.

c. What is asked in the problem?

The problem asks for the perimeter of the trapezoidal frame.

d. What are the given facts?

The given facts are the lengths of the sides of the trapezoidal frame: 60 cm, 65 cm, 80 cm, and 95 cm.