Quadrilateral ABCD: Finding Side DC's Length

Hey math enthusiasts! Let's dive into a geometry problem involving a quadrilateral ABCD. We're given some side lengths in terms of 'x' and the perimeter. Our mission? To figure out the length of side DC. Get ready to flex those math muscles! We'll break down the problem step-by-step, making it super easy to understand. So, grab your pencils and let's get started. This is going to be a fun journey, and by the end, you'll be a quadrilateral whiz. This problem is a classic example of how algebra and geometry work together, and it's a fantastic way to sharpen your problem-solving skills. Whether you're a student, a math lover, or just someone who enjoys a good puzzle, this is for you. We'll use the given information to create an equation, solve for 'x', and then use that value to find the length of DC. It’s like a treasure hunt, but instead of gold, we’re after the perfect solution! Ready to unlock the secrets of this quadrilateral? Let’s jump in!

Understanding the Quadrilateral and the Given Information

Quadrilateral ABCD is a four-sided polygon, a shape with four sides and four angles. In this specific problem, we have a quadrilateral where the lengths of the sides are expressed in terms of 'x'. This is a common setup in algebra, where we use variables to represent unknown quantities. The problem provides us with the following information:

• Side AB = x + 1 cm
• Side BC = x - 1 cm
• Side CD = 2x cm
• Side DA = 2x - 5 cm

We're also given a crucial piece of information: the perimeter of the quadrilateral ABCD is 52 centimeters. Remember, the perimeter of any polygon is the total length of all its sides added together. This is our key to solving the problem. The perimeter gives us a direct relationship between all the side lengths. So, using this information, we'll construct an equation.

Now, let's break down each piece of information. The sides are given as expressions of 'x'. This means that each side's length changes depending on the value of 'x'. The perimeter, on the other hand, is a fixed value, which allows us to find 'x'. The goal is to first find 'x' and then use it to calculate the length of DC. Knowing the perimeter is like having the final piece of a puzzle; it connects all the other pieces (the side lengths) together. Understanding these basics is critical before jumping into the calculations. So, take a moment to absorb these key details, and then we'll move on to the next step: formulating our equation.

Formulating the Equation for the Perimeter

Alright, guys, here’s where the real fun begins: creating the equation! Since we know the perimeter of the quadrilateral is 52 cm, we can set up an equation by adding up all the side lengths and setting the sum equal to 52. Remember, the perimeter is the total distance around the shape. Let's do it step by step. We've got:

• AB = x + 1
• BC = x - 1
• CD = 2x
• DA = 2x - 5

Adding these up, we get: (x + 1) + (x - 1) + (2x) + (2x - 5) = 52. Let's simplify this equation. Combining like terms (the terms with 'x' and the constant numbers), we get: x + x + 2x + 2x + 1 - 1 - 5 = 52. This simplifies to 6x - 5 = 52. Great job, guys! You've successfully translated the problem's information into a solvable algebraic equation. This step is super important because it's the foundation of the rest of the solution.

The equation we have now, 6x - 5 = 52, directly links the side lengths to the known perimeter. Remember that the equation reflects the total length around the shape. Now that we have our equation, we need to solve for 'x'. Solving for 'x' will give us the numerical value that we can plug back into the expressions for each side length. This allows us to find the lengths of all the sides. So, the equation is not only a mathematical statement but also a roadmap that guides us to the solution. Understanding how to set up this equation correctly is key to solving the problem, and you’ve nailed it!

Solving for 'x'

Now that we've set up our equation, it's time to find the value of 'x'! Remember, our equation is 6x - 5 = 52. Our goal here is to isolate 'x' on one side of the equation. To do this, we'll follow these steps:

1. Add 5 to both sides: This will get rid of the -5 on the left side. So, the equation becomes 6x - 5 + 5 = 52 + 5, which simplifies to 6x = 57.
2. Divide both sides by 6: This isolates 'x'. We get 6x / 6 = 57 / 6, which simplifies to x = 9.5.

Ta-da! We've found that x = 9.5. This means that if we plug 9.5 into the expressions for each side length, we'll find the actual lengths of the sides in centimeters. Finding the value of 'x' is like unlocking the secret code to the problem. Each step is designed to simplify the equation and get us closer to our goal. Remember, the balance of the equation is essential. Whatever you do on one side, you must do on the other. That keeps everything fair and square, mathematically speaking!

Now, let's take a quick look back at what we've done so far. We started with the basic information about the quadrilateral, then we formed an equation based on the perimeter. We've now successfully solved for 'x'. It's always a good idea to double-check your work, so take a moment to confirm that your calculations are correct. The next step is to use the value of 'x' to calculate the length of side DC. This is where it all comes together! So, take a deep breath, you're almost there. Solving for 'x' is a huge accomplishment, and you're doing awesome!

Calculating the Length of DC

Alright, folks, we're on the home stretch! We know that side CD = 2x, and we've found that x = 9.5. Now, all we need to do is plug in the value of 'x' to find the length of DC. So, CD = 2 * 9.5 = 19 cm.

Therefore, the length of side DC is 19 centimeters. Boom! You've just solved the problem. Congratulations! This is the grand finale, where everything we've worked on comes together. You took the given information, set up an equation, solved for 'x', and now you've found the length of DC. The feeling of reaching the right answer is fantastic, isn't it? It shows how a little bit of knowledge and a few steps can take you to a successful conclusion. Also, always remember to include the units of measurement (cm in this case) in your final answer. It makes the result complete and clear.

Always double-check your work to avoid small errors. This will help you to build confidence in your problem-solving abilities. Every step you take, from understanding the problem to finding the solution, builds your math skills and your confidence. Keep up the excellent work! You’ve shown that you can break down a complex problem into manageable pieces, which is a great skill in both math and life. Now, you can use these skills to approach other geometry problems with confidence.

Conclusion

So, there you have it, guys! We've successfully calculated the length of side DC in quadrilateral ABCD. We started with the basics, used the perimeter, crafted an equation, solved for 'x', and finally, found that the length of DC is 19 cm. I hope you've enjoyed the journey as much as I have. Always remember that mathematics is about understanding the steps and applying them in the right order. Practicing these kinds of problems helps you understand mathematical concepts better. Also, don't be afraid to ask questions; it's the best way to learn and grow. You did an amazing job today. High five!

To recap:

• Understanding the Problem: We were given the side lengths of a quadrilateral in terms of 'x' and the perimeter.
• Setting up the Equation: We used the perimeter to create an equation: (x + 1) + (x - 1) + 2x + (2x - 5) = 52.
• Solving for 'x': We simplified and solved the equation to find x = 9.5.
• Finding the Length of DC: We used x = 9.5 to calculate DC = 2x = 19 cm.

Well done, everyone! Keep practicing, and you'll become math wizards in no time!