Unlocking Rectangle Perimeters: A Step-by-Step Guide

Hey guys! Let's dive into a fun geometry problem. We're gonna break down how to find the perimeter of a rectangle, and I promise, it's easier than you think! This is all about understanding the shapes around us and how to measure them. So, in this article, we'll talk about how to solve a geometry problem, specifically about the perimeter of a rectangle. We will start with a rectangle, then break it down into smaller parts, and finally, find the expression to calculate its perimeter. We will use the given information about the sides of the rectangle to find our result. Let's get started, shall we?

Understanding the Problem: The Rectangle's Anatomy

Alright, so imagine a rectangle, the classic shape with four sides and four right angles. Now, this particular rectangle [ABFE] is a bit special because it's made up of two parts: a square [ABCD] and another rectangle [EDCF]. We're given some key information to work with: $\overline{EF} = x$ and $\overline{CF} = \frac{x}{3}$. $\overline{EF}$ is the same length as $\overline{AB}$, and $\overline{CF}$ is the same length as $\overline{AD}$. Remember, the perimeter is simply the total distance around the outside of the shape. To find the perimeter of the rectangle [ABFE], we need to add up the lengths of all its sides. This means we'll add $\overline{AB}$, $\overline{BF}$, $\overline{FE}$, and $\overline{EA}$. But wait, how do we know the lengths of all these sides? Well, that's where the given information comes in handy, and we'll use our knowledge of squares and rectangles to figure it out. We will use the information given to us, and with the help of math, we will find the perimeter. This is a journey that many people face during school, so it is important to understand the concept.

Let's break down the sides of the rectangle [ABFE]. We know $\overline{EF} = x$. Since [ABCD] is a square, all its sides are equal. So, $\overline{AB} = \overline{CD} = \overline{BC} = \overline{DA}$. Also, we know that $\overline{CF} = \frac{x}{3}$. Because $\overline{BF} = \overline{BC} + \overline{CF}$, we can find $\overline{BF}$ by adding $\overline{BC}$ and $\overline{CF}$. Now, we know $\overline{EF} = x$, because it is given. We also know that $\overline{AB} = x$, since [ABCD] is a square. So, basically, we have the lengths of all sides. Therefore, the perimeter of the rectangle [ABFE] can be calculated by adding all the sides. Understanding this is super important. It builds a solid foundation for more complex geometry problems later on. Remember, rectangles and squares are everywhere. Being able to quickly calculate their perimeters is a handy skill.

Now, let's visualize this. Imagine the rectangle [ABFE] in front of you. You can see the side $\overline{EF}$ is equal to x. Because $\overline{ABFE}$ is made of a square and a rectangle, we know the top side $\overline{AB}$ is also x. The side $\overline{CF}$ is given as $\frac{x}{3}$. Because $\overline{CF}$ is a part of the side $\overline{BF}$, and $\overline{BC} = x$, we can figure out the entire length of side $\overline{BF}$. See, geometry isn't that hard. It's all about looking at the pieces of the puzzle and putting them together. We will start with the expression for the perimeter, then simplify the expression and we're done. Just take it one step at a time, and you'll be a geometry pro in no time.

The Perimeter Equation: Building the Expression

Okay, time to get to the core of the problem: writing an expression for the perimeter. The perimeter of a rectangle is calculated by adding the lengths of all its sides. So, for the rectangle [ABFE], the perimeter (P) is:

P = AB + BF + FE + EA

We know that $\overline{AB} = x$ and $\overline{FE} = x$. We need to find $\overline{BF}$ and $\overline{EA}$. Now, $\overline{BF}$ is made up of two parts: $\overline{BC}$ and $\overline{CF}$. We know $\overline{CF} = \frac{x}{3}$. Also, since [ABCD] is a square, $\overline{BC} = \overline{AB} = x$. Therefore, $\overline{BF} = x + \frac{x}{3}$. Also, because $\overline{EA}$ is the same length as $\overline{BF}$, $\overline{EA} = x + \frac{x}{3}$.

So now we can rewrite the perimeter equation with what we know: P = x + (x + \frac{x}{3}) + x + (x + \frac{x}{3})

This is our starting expression. Now, it's all about simplifying this expression to make it easier to work with. Remember, we are looking for a simplified expression. This is like tidying up your room – you're just putting things in a more organized way. The same thing can be done with math equations. The steps are very straightforward. Just be careful when performing the operations. That is the only thing you have to do to solve the problem. So, let's simplify it! We can group all the x terms together and the fraction terms together. This makes it easier to work with them.

Simplifying the Expression: Putting it all Together

Alright, let's simplify the expression. We have: P = x + (x + \frac{x}{3}) + x + (x + \frac{x}{3})

First, let's combine the x terms. We have x + x + x + x, which equals 4x. Next, let's combine the fraction terms: \frac{x}{3} + \frac{x}{3}, which equals \frac{2x}{3}. So, now we have:

P = 4x + \frac{2x}{3}

We can simplify this further by combining the terms. To do this, we need to find a common denominator, which is 3. We can rewrite 4x as \frac{12x}{3}. So, the expression becomes:

P = \frac{12x}{3} + \frac{2x}{3}

Now, we can add the numerators:

P = \frac{14x}{3}

And there you have it! The simplified expression for the perimeter of the rectangle [ABFE] is \frac{14x}{3}. We have successfully simplified the expression. From the beginning, we have the original expression, which is not simplified. Then, we simplify the expression by combining similar terms. Finally, we make it into the most simplified expression. Easy, right?

This whole process, guys, is all about taking a complex shape and breaking it down into smaller, manageable parts. We used our knowledge of squares, rectangles, and basic algebra to solve the problem. Practice this, and you'll be a pro in no time.

In summary:

• We understood the problem by analyzing the rectangle [ABFE] and its components.
• We wrote the perimeter equation, using the given information.
• We simplified the expression by combining terms and finding a common denominator.
• We found the simplified expression for the perimeter of the rectangle. With enough practice, you'll become confident in solving geometry problems and understanding shapes. Keep up the good work!

This problem-solving approach is not just for math; it can be applied to many aspects of life! Great job!