Finding The Largest Side Of A Rectangle: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a geometry problem that's super common in exams and real-life scenarios. We'll be figuring out the largest side of a rectangle, armed with just its perimeter and a relationship between its sides. So, grab your pencils, and let's get started. This guide will walk you through the problem step-by-step, making sure you understand the logic and can apply it to similar challenges. We'll break down the concepts, use clear language, and avoid jargon, so everyone can follow along. No need to be a math whiz – just a little curiosity is enough!

Understanding the Problem: The Basics of Rectangles and Perimeter

Alright, let's unpack the problem. We're given a rectangle, and we know its perimeter is 30 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 cover. We're also told that one side of the rectangle is 2/3 the length of the other side. This is crucial information that we'll use to crack the code. The question asks us to determine the measure of the largest side of this rectangle, and we need to choose from the following options: A) 10 cm, B) 9 cm, C) 8 cm, D) 7 cm, or E) 6 cm. This looks like a straightforward geometry problem, right?

Before we jump into the calculations, let's make sure we're on the same page about rectangles. A rectangle is a four-sided shape (a quadrilateral) where all four angles are right angles (90 degrees). Opposite sides of a rectangle are equal in length. This is a key property we'll use. So, if we call the length of one side 'l' and the length of the other side 'w' (for width), we know that the other two sides will also have lengths 'l' and 'w', respectively. The perimeter (P) of a rectangle is calculated using the formula: P = 2l + 2w. This formula simply adds up the lengths of all four sides. In our problem, we know the perimeter (30 cm), and we have a relationship between the length and width (one is 2/3 of the other). The main goal is to use this information to determine the value of the largest side of the rectangle. Are you ready to dive deeper?

Setting Up the Equations: Translating Words into Math

Now, let's translate the problem into mathematical equations. This is where the real problem-solving begins! We've already established the basics, and now it's time to put our knowledge into action. The first piece of information we have is the perimeter of the rectangle, which is 30 cm. Using the perimeter formula, we can write: 2l + 2w = 30. This equation tells us the sum of all the sides equals 30 cm. Remember, 'l' represents the length, and 'w' represents the width. The second piece of information tells us that one side is 2/3 of the other side. Let's assume that the width (w) is the shorter side. Then, we can write: w = (2/3)l. This equation expresses the relationship between the width and length. Now we have two equations: 1) 2l + 2w = 30 and 2) w = (2/3)l. This is a system of equations. Our job is to solve this system to find the values of 'l' and 'w'. Once we have those values, we'll easily identify the larger side. Are you ready to see how it works?

To solve this, we can use a method called substitution. We'll take the value of 'w' from the second equation and substitute it into the first equation. This will give us an equation with only one variable ('l'), which we can then solve. By understanding this process, we'll be able to solve similar problems. It is really important to grasp these fundamentals to effectively approach mathematical challenges. Let's see how this works: By substitution, the first equation (2l + 2w = 30) becomes 2l + 2*(2/3)l = 30. Notice how we replaced 'w' with '(2/3)l'. Now, let's simplify and solve for 'l'. First, multiply: 2l + (4/3)l = 30. Then, combine the terms with 'l'. You need to find a common denominator to add 2l and (4/3)l. Remember that 2l can be written as (6/3)l. So, (6/3)l + (4/3)l = 30, which simplifies to (10/3)l = 30. To find 'l', multiply both sides of the equation by (3/10): l = 30 * (3/10). This gives us l = 9. So, the length (l) of the rectangle is 9 cm. This also means, if we're working in the opposite direction, we'll find the value of w.

Solving for the Length and Width: Finding the Sides of the Rectangle

Great job on setting up the equations and understanding the substitution method! Now, let's put it all together to find the actual dimensions of our rectangle. We've already determined that the length (l) is 9 cm. Now, we'll use the relationship between length and width to find the width (w). Remember our equation: w = (2/3)l. We know 'l' is 9 cm, so we can substitute that into the equation: w = (2/3) * 9. Calculate it to find the width. w = 6 cm. So, the width (w) of the rectangle is 6 cm. At this point, we've found both the length and width of the rectangle: length (l) = 9 cm and width (w) = 6 cm. Now that we know the dimensions, the final step is to determine the largest side of the rectangle. Given that the sides are 9 cm and 6 cm, it's clear that the largest side is 9 cm. The question asked us to determine the measure of the largest side of this rectangle. By performing these calculations, we've arrived at the solution.

This methodical approach breaks down the problem, making it easier to solve. When you encounter similar geometry problems, remember the importance of setting up equations based on the information provided, using the perimeter formula and applying the relationship between sides. Once you've found the dimensions, identifying the largest side is a piece of cake. This whole process is more about understanding the concepts than memorizing formulas. Congratulations, guys. You've successfully solved the problem! Let's check our answer against the options.

Determining the Correct Answer: Checking Our Work and Choosing the Right Option

Now, let's review our findings and select the correct answer. We found that the length of the rectangle is 9 cm, and the width is 6 cm. The question asks for the measure of the largest side. Comparing the two dimensions, we see that 9 cm is larger than 6 cm. Therefore, the largest side of the rectangle is 9 cm. Now, let's go back to the multiple-choice options and identify the one that matches our answer. The options were: A) 10 cm, B) 9 cm, C) 8 cm, D) 7 cm, E) 6 cm. Our calculated answer, 9 cm, matches option B. Thus, the correct answer is B) 9 cm. Congratulations! You've successfully solved the problem and found the correct answer. This entire process demonstrates a clear understanding of geometry principles and problem-solving techniques. By breaking down the problem step-by-step, we've made the solution clear and accessible. Always remember to check your answer against the given options to ensure accuracy. If your calculated answer isn't among the choices, revisit your steps to catch any errors. If you're struggling to understand a problem, it might be beneficial to review the problem-solving strategies, such as the substitution method or the perimeter formula.

Conclusion: Mastering the Rectangle Problem

Awesome work, everyone! We've successfully navigated the world of rectangles and perimeter. By understanding the relationships between sides and applying the perimeter formula, we were able to solve the problem systematically. Remember, the key is to break down complex problems into smaller, manageable steps. In this example, we started by understanding the problem, then formulated equations based on the given information. Then, we used the substitution method to solve for the unknown side lengths, and finally, we selected the appropriate answer from the given options. Math can be so much fun.

Mastering these skills will empower you to tackle a wide variety of geometry problems. Don't hesitate to practice more problems to sharpen your skills. The more you practice, the more confident you'll become! So, keep exploring the world of mathematics, and never stop learning. Each problem you solve builds your confidence and strengthens your problem-solving abilities. Stay curious, keep practicing, and remember that every step is a step towards math mastery. Keep up the great work, and you'll be acing these geometry problems in no time! Remember, understanding the principles is far more valuable than memorizing formulas.

Now you've got this! If you have any questions or want to try another problem, feel free to ask. Keep up the great work, and happy learning! Remember, understanding the principles is far more valuable than memorizing formulas. Keep practicing and keep exploring the amazing world of mathematics!