Solve Rectangle Area Problem A Step-by-Step Guide

Hey everyone! Let's dive into a fun geometry problem that involves a rectangular piece of paper, some clever cuts, and a bit of math magic. We're going to explore how to find the dimensions and area of this rectangle using the information given. So, grab your thinking caps, and let's get started!

Problem Breakdown: Unraveling the Rectangular Mystery

Let’s first understand the problem. We have a rectangular piece of paper where the width is 3 inches less than the length. Imagine this rectangle in your mind – it's longer than it is wide. Now, we make a cut along the diagonal, splitting the rectangle into two identical right-angled triangles. Each of these triangles has an area of 44 square inches. Our mission, should we choose to accept it (and we do!), is to figure out which statements about this rectangle are true.

This problem is a fantastic blend of geometry and algebra. We'll need to use our knowledge of rectangles, triangles, and how their areas are calculated. We'll also be setting up some equations to solve for the unknown dimensions. Don't worry, it's not as daunting as it sounds! We'll break it down step by step, making sure everyone can follow along.

Remember, the key to solving any math problem is understanding what it's asking. So, let's keep the picture of the rectangle and the triangles in our minds as we move forward. We'll be piecing together the information like detectives solving a case, using each clue to get closer to the solution. Are you ready to put on your detective hats? Let's go!

Setting Up the Equations: Translating Words into Math

Alright, time to put our algebraic skills to the test! In these types of geometry problems, translating the words into mathematical equations is key. It's like learning a new language, but instead of words, we're using symbols and numbers. So, let's decode the given information and turn it into equations we can actually work with.

We know that the width of the rectangle is 3 inches less than its length. If we let 'l' represent the length and 'w' represent the width, we can write this as: w = l - 3. See? We've already turned a sentence into a neat little equation. This is our first piece of the puzzle. Think of it as the foundation upon which we'll build our solution. It tells us the relationship between the length and the width, and that's super important.

Next up, we have the triangles. We know that cutting the rectangle along the diagonal creates two congruent (identical) right triangles, each with an area of 44 square inches. Remember how to find the area of a triangle? It's half the base times the height. In our case, the base and height of each triangle are the length and width of the rectangle. So, the area of one triangle can be written as: (1/2) * l * w = 44. This is our second equation, and it links the length and width to the area of the triangles. We're on a roll!

Now we have two equations: w = l - 3 and (1/2) * l * w = 44. This is a system of equations, and it's our roadmap to finding the values of 'l' and 'w'. We'll use these equations to solve for the length and width, and once we have those, we can answer any question about the rectangle. It's like having the secret code to unlock the puzzle! So, let's move on to the next step: solving these equations. We're getting closer to the solution, guys!

Solving for Length and Width: Cracking the Code

Okay, detectives, it's time to put our equation-solving hats on! We have two equations and two unknowns, which means we're in business. Our goal here is to find the values of 'l' (length) and 'w' (width). There are a few ways we can tackle this, but one common method is substitution. It's like swapping one ingredient for another in a recipe to get the same delicious result.

Remember our equations? We have w = l - 3 and (1/2) * l * w = 44. The first equation, w = l - 3, is perfect for substitution because it tells us exactly what 'w' is in terms of 'l'. So, we're going to take that expression for 'w' (which is l - 3) and plug it into the second equation wherever we see a 'w'. This will give us one equation with just one variable, 'l', which is much easier to solve.

So, let's do it! Substituting 'l - 3' for 'w' in the second equation, we get: (1/2) * l * (l - 3) = 44. Now we have a single equation with just 'l', and it's a quadratic equation! Don't worry, we can handle this. First, let's get rid of the fraction by multiplying both sides of the equation by 2: l * (l - 3) = 88. Next, we distribute the 'l': l^2 - 3l = 88.

To solve a quadratic equation, we want to set it equal to zero. So, we subtract 88 from both sides: l^2 - 3l - 88 = 0. Now we have a classic quadratic equation in the form of ax^2 + bx + c = 0. We can solve this by factoring, using the quadratic formula, or even completing the square. Factoring is often the quickest if we can spot the factors. Can you think of two numbers that multiply to -88 and add up to -3? Give it a try!

Once we find those factors, we can rewrite the quadratic equation in factored form and solve for 'l'. Remember, 'l' represents the length, so it has to be a positive number. Once we have 'l', we can easily find 'w' using the equation w = l - 3. We're like codebreakers at this point, cracking the mathematical code to reveal the dimensions of our rectangle. Let's keep going, we're almost there!

Calculating the Area: Putting It All Together

Alright, we've done the hard work of solving for the length and width. Now comes the fun part: using those values to calculate the area of the original rectangular piece of paper. This is where everything comes together, and we see the final result of our mathematical journey.

Remember, the area of a rectangle is simply its length multiplied by its width: Area = l * w. We've already found 'l' and 'w' by solving our system of equations. So, all that's left to do is plug those values into the formula and do the multiplication. It's like the grand finale of a fireworks show, where all the pieces come together to create a dazzling display.

But wait, before we jump straight into the calculation, let's take a moment to think about what we expect the area to be. We know that the rectangle was cut into two triangles, each with an area of 44 square inches. So, the total area of the rectangle should be the sum of the areas of the two triangles. Can you figure out what that is? It's a simple addition, but it's a great way to check our work. If our calculated area doesn't match this total, we know we need to go back and look for a mistake.

Once we've done the multiplication and found the area, we'll have a complete picture of our rectangle. We'll know its length, its width, and its total area. We'll have solved the mystery of the paper puzzle! And that's a pretty satisfying feeling, guys. So, let's get those calculators out (or do the math by hand, if you're feeling old-school) and find the area. We're crossing the finish line!

True Statements: Verifying the Solution

We've reached the final stage of our problem-solving adventure! We've calculated the length, width, and area of the rectangle. Now, the last step is to verify which statements about the rectangle are true. This is like checking our answers on a test to make sure we've got everything right.

Remember the original problem asked us to check all the statements that apply. This means there could be one true statement, multiple true statements, or even none! We need to carefully examine each statement and see if it matches the information we've uncovered about the rectangle.

To do this, we'll take each statement and treat it like a question. Does the statement accurately describe our rectangle? If it does, we check it off. If it doesn't, we leave it unchecked. It's like being a detective again, comparing the evidence (our calculations) to the claims (the statements) and deciding which ones hold up.

This step is super important because it ensures we've fully understood the problem and haven't made any mistakes along the way. It's not enough to just find the length, width, and area; we need to be able to use that information to answer specific questions about the rectangle. So, let's put on our critical-thinking caps and carefully evaluate each statement. We're in the home stretch now, guys! Let's make sure we finish strong and get all the right answers.

By following these steps, we can confidently solve this geometry problem and learn some valuable math skills along the way. Remember, math isn't just about numbers and equations; it's about problem-solving, critical thinking, and the satisfaction of finding a solution. So, keep practicing, keep exploring, and keep having fun with math!