If A Square's Sides Increase By 5 Cm, Its Area Becomes 100 Cm². (a) If The First Square's Side Is X, What Is The Second Square's Side Length? (b) Find The Side Lengths Of Both Squares.

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In the fascinating realm of geometry, squares stand as fundamental shapes, their symmetrical nature and predictable properties making them a cornerstone of mathematical understanding. This article delves into an intriguing problem involving squares, where an increase in side length leads to a change in area. We'll unravel the relationships between side length, area, and how these properties interact when we modify the dimensions of a square.

Deciphering the Square's Transformation: A Step-by-Step Exploration

Let's embark on a journey to decipher the mysteries of square transformations. We'll begin by laying out the problem statement and then systematically dissecting it to arrive at a comprehensive solution. The core of our investigation lies in understanding how changes in a square's side length directly impact its area. This principle is crucial in various fields, from architecture and engineering to computer graphics and design.

(a) Expressing the Side Length of the Enlarged Square

Let's start by defining the variables and setting up the framework for our solution. We're given that the original square has sides of length 'x' cm. This forms the foundation upon which our calculations will be built. Now, imagine increasing each side of this square by 5 cm. This enlargement naturally alters the square's overall dimensions. To represent this change mathematically, we need to express the new side length in terms of our original variable, 'x'. This simple yet powerful step allows us to connect the initial state of the square to its transformed state, paving the way for further analysis.

The side length of the enlarged square is simply the original side length plus the increase. Therefore, the sides of the second square measure (x + 5) cm. This expression is the key to unlocking the relationship between the original square and its enlarged counterpart. It allows us to directly compare their dimensions and, ultimately, their areas. This representation is crucial for understanding how geometric shapes change when their parameters are altered.

(b) Unveiling the Side Lengths: A Quest for Numerical Solutions

Now comes the exciting part: finding the actual side lengths of our squares. We know that the area of the enlarged square is 100 cm². This piece of information is the crucial link that allows us to translate our symbolic representation into a concrete numerical solution. The area of a square, as we know, is simply the square of its side length. Therefore, we can set up an equation that relates the side length of the enlarged square, which we've expressed as (x + 5) cm, to its area, 100 cm². Solving this equation will reveal the value of 'x', the side length of our original square, and subsequently, the side length of the enlarged square.

To find the length of the sides, we'll use the information about the area of the enlarged square. We know that area of a square = (side length)². Since the area of the enlarged square is 100 cm², we can write the equation:

(x + 5)² = 100

This equation encapsulates the essence of our problem. It mathematically connects the unknown side length, 'x', to the known area of the enlarged square. Solving this equation is a critical step in understanding the geometric relationship between the squares.

To solve for x, we first take the square root of both sides of the equation:

√(x + 5)² = √100

This simplifies to:

x + 5 = ±10

Note that we consider both positive and negative roots when taking the square root. This is a crucial step in ensuring we capture all possible solutions. However, in the context of our problem, we're dealing with physical lengths, which cannot be negative. Therefore, we'll focus on the positive solution.

We now have two possible equations:

  • x + 5 = 10
  • x + 5 = -10

Since side lengths cannot be negative, we discard the second equation (x + 5 = -10) as it would lead to a negative value for x. This is a critical step in ensuring our solution aligns with the physical constraints of the problem.

Solving the first equation, x + 5 = 10, we subtract 5 from both sides:

x = 10 - 5

This gives us:

x = 5

Therefore, the length of the sides of the first square is 5 cm. This is a significant finding, as it provides the foundation for understanding the dimensions of the original square.

Now, to find the length of the sides of the second square, we use the expression we derived earlier, (x + 5) cm. Substituting x = 5 cm, we get:

Side length of the second square = 5 + 5 = 10 cm

This completes our quest. We've successfully determined the side lengths of both squares, unveiling the mystery of their dimensions. The original square has sides of 5 cm, while the enlarged square boasts sides of 10 cm. This numerical solution provides a tangible understanding of the problem and reinforces the relationship between side length and area in squares.

Conclusion: Embracing Geometric Insights

In conclusion, by carefully analyzing the problem and applying fundamental geometric principles, we've successfully determined the side lengths of both squares. The original square has sides of 5 cm, while the enlarged square, with an area of 100 cm², has sides of 10 cm. This exploration highlights the power of mathematical reasoning in unraveling geometric relationships. Understanding how changes in side length affect area is a crucial concept in various fields, reinforcing the importance of geometric principles in our everyday world. This exercise not only strengthens our problem-solving skills but also deepens our appreciation for the elegance and precision of mathematics.