Solving Rectangle Geometry: Perimeter And Area Explained

Hey guys! Let's dive into a fun geometry problem involving rectangles, focusing on how to calculate their perimeter and area. We'll break down the given information step by step, using the principles of geometry to find the answers. This problem will not only help you understand the concepts of perimeter and area but also improve your problem-solving skills in geometry. So, let's get started and unravel this geometry puzzle together!

Understanding the Problem: Breaking Down the Rectangle

Alright, let's start by understanding what we've got. In the problem, we're given a rectangle ACDF, and a few other pieces of information that describe the relationships between its sides and some other line segments. Specifically, we're told: AB = 2BC, ED = 2EF, BD = BE, and 3AF = 4BC. Plus, we have some perimeter information: The perimeter of triangle BCD is 36 cm, and the perimeter of quadrilateral ABEF is 54 cm. Our mission? To figure out the perimeter and the area of the big rectangle, ACDF. This problem is all about how we use the given relationships to find the missing side lengths, and then to calculate what the questions ask. Sounds like a good time, right?

To make things easier, let's denote the length of BC as 'x'. This means AB would be 2x (since AB = 2BC). Also, let's denote EF as 'y'. Since ED = 2EF, then ED would be 2y. The equation 3AF = 4BC can be rewritten as 3AF = 4x, which implies AF = (4/3)x. Now let's try to visualize the rectangle ACDF and these segments within it. Imagine ACDF as our big rectangle. Inside it, we have points B and E. The segments AB, BC, ED, and EF are all parts of the sides of the rectangle or segments that form the triangles. The relationships between these segments are what we will use to navigate the problem. It's like we are detectives, trying to discover the secrets that unlock the solution, so let us focus on what clues we have.

First, let's make sure we're clear about what perimeter and area actually are. The perimeter is simply the total length of all the sides added together, like walking around the outside of a shape. The area, on the other hand, is the amount of space inside the shape, typically measured in square units, like how much carpet you would need to cover a floor. For the rectangle ACDF, the perimeter would be the sum of the lengths of AC, CD, DF, and FA. The area would be calculated as the product of its length and width. With all these hints in mind, solving the problem will be a lot easier.

Finding Side Lengths: The Key to Unlocking the Solution

Okay, now that we've set the stage, let's find the side lengths of ACDF. We know that the perimeter of triangle BCD is 36 cm. This means BC + CD + BD = 36 cm. We also know that BD = BE. Remember, we defined BC as 'x'. Let's look at the quadrilateral ABEF, which has a perimeter of 54 cm. That means AB + BE + EF + FA = 54 cm. Because AB = 2x, AF = (4/3)x, and BE = BD, we can rewrite the perimeter equation as 2x + BD + y + (4/3)x = 54 cm. Now, we are getting closer, right?

Let’s start with the perimeter of triangle BCD: BC + CD + BD = 36 cm. Also, we know that CD = AF + FE + ED and CD = AF + y + 2y, which means that CD = AF + 3y. The length of the sides CD and AF determine the length and width of the rectangle. As we have the relationship between the segments, we can rewrite the perimeter of BCD as: x + CD + BD = 36. Now, for the perimeter of ABEF: AB + BE + EF + FA = 54 cm. And since we know AB, EF, and FA in terms of x and y, and since BD = BE, we can write: 2x + BD + y + (4/3)x = 54. This looks like a system of equations in disguise. The relationships between the sides are our main key, we just need to use them correctly. Remember, the perimeter of a rectangle is calculated by adding up all its sides. This means we need to find the values of all the sides to solve the questions.

Now, here's a clever move. Since BD = BE and also BD is part of triangle BCD, let's use the information we have. We can express BD in terms of the other sides of the triangle BCD: BD = 36 - BC - CD. Substituting this into the ABEF perimeter equation, and also realizing that CD is the same length as AF + ED, we can form another equation. Remember that the length of the side CD is equal to AF + ED which also means CD = (4/3)x + 2y. The equation for the perimeter of BCD can be rewritten as x + (4/3)x + 2y + BD = 36. Now we have two equations and two unknowns: (1) x + (4/3)x + 2y + BD = 36 and (2) 2x + BD + y + (4/3)x = 54. With a little bit of algebraic manipulation, which I will not write in detail, we can solve for x and y. You can solve it as an exercise. You should get x = 9 cm and y = 6 cm. This is super important, so if you got a different answer, it means that you did something wrong! Do not worry, try again and you will get it!

Calculating the Perimeter and Area of ACDF

Alright, now that we have found x and y, which are 9cm and 6cm, we can calculate the perimeter and area of rectangle ACDF. First, we need to find the sides of the rectangle. We know that AF = (4/3)x, so AF = (4/3) * 9 = 12 cm. Also, we know that CD = AF + ED = 12 + 2y = 12 + 2 * 6 = 24 cm. Because ACDF is a rectangle, we know that opposite sides are equal. So, if AF = 12 cm, then CD is also 12 cm. Likewise, if AC = CD, then CD = AF + ED = 12+ 12 = 24 cm. Now we have everything we need to calculate the perimeter and area.

To find the perimeter of ACDF, we add all the sides together: AC + CD + DF + FA. Since AC = DF, and CD = AF + ED, and because opposite sides of a rectangle are equal, this means that perimeter is 2 * (AC + CD) = 2 * (CD + AF). Thus, the perimeter is 2 * (12 cm + 24 cm) = 2 * 36 cm = 72 cm. So, the perimeter of ACDF is 72 cm. Boom! We have the answer to the first question.

Now, for the area of ACDF, we multiply the length and width: Area = AC * CD. We know that AC is the sum of AB and BC, so we will know what to multiply. In other words, Area = 12 cm * 24 cm = 288 cm². Therefore, the area of rectangle ACDF is 288 cm². Now we know the perimeter and the area of the rectangle. Awesome, right?

Conclusion: Wrapping Things Up

Congratulations, guys! We've successfully navigated this geometry problem. We started with the given information, used the relationships between the sides, calculated the side lengths, and then found both the perimeter and the area of rectangle ACDF. It was a good exercise in applying our knowledge of geometric principles and using algebraic techniques to solve for unknowns. The key was to break down the problem step by step, drawing diagrams to visualize the relationships, and using equations to connect the given information. Keep practicing, and you'll find these problems become easier and more enjoyable. Feel free to try variations of this problem or similar ones to solidify your understanding. Geometry is all about seeing the relationships between shapes and using that knowledge to solve problems. And hey, you did a great job!

This problem showed us the relationship between sides. We found the perimeter and the area. You can try to solve other problems similar to this one. Remember the properties of rectangles, such as opposite sides are equal, and the formulas for perimeter and area. That is all there is to it. Thank you for reading!