Rectangle Area Problem: Find Shaded Region's Area

Hey guys! Let's dive into a cool geometry problem involving rectangles and areas. This one's a classic that often pops up in math discussions, especially for middle school levels. We're going to break it down step-by-step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so here's the deal. We've got a rectangle ABCD, and there's a point E hanging out right in the middle of side AD. Then we have another point, F, somewhere on DC. The tricky part? The ratio of CF to FD is 3:2. We also know that side AB is 5√3 cm long, and the whole rectangle's area is 216 cm². Our mission, should we choose to accept it, is to find the area of the shaded region in the figure. Sounds like a puzzle, right? But don't worry, we'll crack it!

This is a typical geometry problem where visualizing and understanding the relationships between different parts of the figure is super crucial. We need to figure out how the given information – the side length, the area, and the ratio – all connect to help us find the area of that shaded region. Think of it like a detective game, where we're using clues to solve the mystery.

To tackle this, we will need to use our knowledge of rectangles, areas, and ratios. Remember, a rectangle's area is just its length times its width. And ratios help us understand how different lengths relate to each other. We will combine these concepts to zoom in on the area we're trying to find. It's like fitting the pieces of a jigsaw puzzle together – each bit of information helps us see the bigger picture. So, let's start piecing things together and see how we can solve this!

Breaking Down the Rectangle

Let's start with the basics: the rectangle ABCD. We know its area is 216 cm², and one of its sides, AB, is 5√3 cm. Remember the formula for the area of a rectangle? It's simply Area = length × width. So, we can use this to figure out the length of the other side, AD. Think of it like this: if we know the total area and one side, we can always find the other side by dividing. It's like reverse engineering the area formula!

So, to find the length of AD, we'll divide the area (216 cm²) by the length of AB (5√3 cm). This might seem a bit tricky with that square root in there, but don't sweat it! We'll handle the math carefully. Dividing 216 by 5√3 might give you a decimal, but let's try to simplify it. We can rationalize the denominator, which basically means getting rid of the square root in the bottom of the fraction. This makes the numbers easier to work with, and it's a neat trick to keep in your math toolkit.

Once we've calculated the length of AD, we'll have a much clearer picture of our rectangle. Knowing both sides AB and AD is key because it gives us the overall dimensions. From there, we can start looking at how the points E and F divide the sides and how that affects the areas within the rectangle. It's all about breaking down the big shape into smaller, more manageable parts. This is a common strategy in geometry – when a problem seems overwhelming, chop it into smaller pieces that you can solve one at a time!

Working with Ratios: CF and FD

Now, let's zoom in on the ratio CF:FD, which is given as 3:2. This ratio is super important because it tells us how the side DC is divided by the point F. Think of it like this: if we imagine DC as being cut into 5 equal parts (3 + 2), then CF takes up 3 of those parts, and FD takes up 2 parts. Understanding this ratio is crucial for figuring out the areas of the smaller triangles within the rectangle.

To really make sense of this, we need to know the total length of DC. Guess what? DC is the same length as AB, which we already know is 5√3 cm. So now, we can actually calculate the lengths of CF and FD. We'll use the ratio to split the total length into the correct proportions. It's like dividing a cake into slices based on a recipe – the ratio tells us how big each slice should be.

Let's do the math: if DC is 5√3 cm and the ratio is 3:2, we can find the length of CF by multiplying the total length by 3/5 (since CF is 3 parts out of the total 5). Similarly, we can find the length of FD by multiplying the total length by 2/5. Once we have these lengths, we're one step closer to figuring out the areas of those triangles. Ratios might seem abstract, but they're a powerful tool for understanding proportions and making calculations in geometry problems. They help us connect different parts of the figure and see how they relate to each other.

Calculating Areas: Triangles and More

Here comes the fun part: calculating areas! We're particularly interested in the areas of triangles because the shaded region is likely made up of triangles or a combination of triangles and other shapes. Remember the basic formula for the area of a triangle: Area = 1/2 × base × height. We'll be using this a lot, so make sure it's fresh in your mind. The trick is to identify the base and height for each triangle we're looking at. Sometimes, it's super obvious, but other times, we might need to do a little bit of thinking to figure out which sides are the base and height.

We have a few key triangles to consider. Think about triangle ADF, triangle CFE, and any other triangles that make up the shaded region. For each triangle, we'll use the lengths we've already calculated (like AD, FD, CF, etc.) to find the base and height. Remember, the base and height need to be perpendicular to each other – that is, they need to form a right angle. This is where knowing the properties of rectangles comes in handy, since rectangles have those nice, neat right angles.

Once we've found the areas of the individual triangles, we might need to add or subtract them to get the area of the shaded region. It's like piecing together the final parts of the puzzle. If the shaded region is made up of multiple shapes, we'll add their areas together. If the shaded region is what's left over after cutting out a shape, we'll subtract the area of the cut-out shape. This is a common technique in geometry problems, where we break down complex shapes into simpler ones, find their areas, and then combine them to get our final answer.

Putting It All Together

Alright, guys, we've done the groundwork, and now it's time to put all the pieces together! We've calculated side lengths, worked with ratios, and found areas of triangles. The final step is to combine all this information to find the area of that elusive shaded region. This is where we really see how all our hard work pays off. It's like the grand finale of our math detective story!

We'll carefully review each step we've taken, making sure we haven't missed anything. We'll double-check our calculations and make sure everything adds up correctly. This is super important to avoid silly mistakes. It's like proofreading a piece of writing – a quick review can catch any typos or errors.

Finally, we'll present our answer in a clear and concise way, including the correct units (cm², since we're dealing with area). We'll also make sure our answer makes sense in the context of the problem. Does it seem like a reasonable size for the shaded region? If it's way too big or way too small, that might be a sign that we've made a mistake somewhere. This final check is a good habit to get into – it helps you build confidence in your answer and makes sure you've really solved the problem. So, let's do this! Let's put it all together and find the area of the shaded region!

Final Answer and Conclusion

After all the calculations and putting the pieces together, the final answer should be one of the options provided (A, B, C, D, or E). Make sure to clearly state which option is the correct one and, if possible, briefly explain why it's the right answer. This shows that you not only arrived at the correct answer but also understand the reasoning behind it. It's like giving a summary of your detective work, showing how you solved the mystery.

So, there you have it! We've successfully tackled a geometry problem involving rectangles, ratios, and areas. We broke it down into manageable steps, used key formulas, and pieced together the information to find the solution. Remember, guys, geometry problems can seem tricky at first, but with a little bit of patience and a systematic approach, you can conquer them! Keep practicing, keep exploring, and you'll become a geometry whiz in no time!

If you found this breakdown helpful, feel free to share it with your friends or classmates who might be struggling with similar problems. And remember, math is like a muscle – the more you use it, the stronger it gets. So, keep flexing those math muscles and challenging yourself with new problems. You got this!