Solving A Geometry Problem: Finding The Square's Side

Hey guys! Let's dive into a cool geometry problem. We've got a square, and we're going to do some cutting and rearranging to create a rectangle. The question is, based on the new rectangle's area, what was the original size of the square? It's a classic example of how understanding area and side lengths can unlock a problem. Let's break it down step-by-step and find the solution. Get ready to flex those math muscles!

Understanding the Problem and Setting Up the Basics

Alright, so here's the deal: We start with a square, which we'll call ABCD. This means all four sides (AB, BC, CD, and DA) are equal in length. Now, we're told that we're going to remove some length from the sides. Specifically, we remove 2 cm from sides AB and DC, and we remove 3 cm from sides AD and BC. The result of these subtractions is a brand-new shape – a rectangle. The area of this rectangle is given to us as 20 cm². Our mission, should we choose to accept it, is to figure out the original side length of the square ABCD. Sounds fun, right?

To get started, let's use some algebra. Let's say the side length of the original square ABCD is 'x' cm. Since all sides of a square are equal, AB = BC = CD = DA = x. Now, let's see what happens when we make the cuts. When we take away 2 cm from AB and DC, the new length of those sides in the rectangle will be (x - 2) cm. Similarly, taking away 3 cm from AD and BC gives us a new length of (x - 3) cm for the sides of the rectangle. Do you see how we're using 'x' as a placeholder to represent the unknown side length of the square? This is super important because it allows us to create equations that we can solve.

The key to solving this problem lies in the area of the rectangle. We know the area of a rectangle is calculated by multiplying its length by its width. In our case, the length is (x - 2) cm, and the width is (x - 3) cm. We are also told that the rectangle's area is 20 cm². So, we can create an equation to represent this relationship: (x - 2) * (x - 3) = 20. This is where the magic happens – we've transformed a geometry problem into an algebraic one!

Formulating the Equation and Solving for X

Okay, so we have our equation: (x - 2) * (x - 3) = 20. Now, let's do some algebra and solve for 'x.' First, we need to expand the expression on the left side of the equation. We use the distributive property (or the FOIL method, if you're familiar with it) to multiply the terms: (x - 2) * (x - 3) becomes x² - 3x - 2x + 6. Simplifying this gives us x² - 5x + 6. So our equation now looks like this: x² - 5x + 6 = 20.

Next, we need to get everything on one side of the equation to set it equal to zero. We do this by subtracting 20 from both sides: x² - 5x + 6 - 20 = 0. This simplifies to x² - 5x - 14 = 0. This is a quadratic equation! There are a few ways to solve this. You can factor the quadratic expression, complete the square, or use the quadratic formula. Let's try factoring.

We need to find two numbers that multiply to -14 (the constant term) and add up to -5 (the coefficient of the x term). Those numbers are -7 and 2, because (-7) * 2 = -14 and (-7) + 2 = -5. Therefore, we can factor the quadratic equation as (x - 7)(x + 2) = 0. Now, for the product of two factors to be zero, at least one of them must be zero. So, we set each factor equal to zero and solve for x. If x - 7 = 0, then x = 7. If x + 2 = 0, then x = -2. We've found two possible solutions for 'x'.

Interpreting the Solution and Finding the Correct Answer

We've crunched the numbers, and now we have two possible answers: x = 7 and x = -2. But wait a minute! We're dealing with a real-world geometry problem. Remember that 'x' represents the side length of a square. Can a side length be negative? Nope! So, we can discard the solution x = -2. It doesn't make sense in the context of the problem.

Therefore, the only valid solution is x = 7 cm. This means the original square ABCD had sides of 7 cm each. Let's double-check our work. If the original side length was 7 cm, then after removing 2 cm from two sides and 3 cm from the other two, the rectangle would have sides of (7 - 2) = 5 cm and (7 - 3) = 4 cm. The area of this rectangle would be 5 cm * 4 cm = 20 cm², which matches the information given in the problem. Success!

So, the answer is (a) because it is not provided, but the side of the square measures 7cm. The square ABCD has a side length of 7 cm. We took a geometry problem, used algebra to create an equation, solved for the unknown, and then interpreted the solution to find our answer. Pretty cool, right?

Conclusion: Putting It All Together

Alright guys, we've successfully navigated a geometry problem and found the side length of the original square. We used algebra, a bit of critical thinking, and a good understanding of area to solve it. Remember, these types of problems are designed to test not just your math skills, but also your ability to apply those skills in a logical way. Always pay attention to the context of the problem and make sure your answer makes sense.

Keep practicing! The more you practice, the better you'll get at recognizing patterns and applying the right techniques to solve different types of problems. Don't be afraid to break problems down into smaller steps, draw diagrams, and check your work. And most importantly, have fun with it! Math can be a blast when you approach it with curiosity and a willingness to learn. Now go out there and tackle some more problems! You've got this!

In summary:

• We were given a square and information about a rectangle created from it.
• We used the area of the rectangle to create an equation.
• We solved the quadratic equation to find the side length of the square (x = 7 cm).
• We validated our answer by making sure it made sense in the context of the problem.