Solving for the Original Side Length of a Square

In this article, we explore a mathematical puzzle involving the side length of a square. By understanding the problem and applying basic algebra and geometry concepts, we can find the original side length of the square. This example will serve as a practical exercise in problem-solving and a reminder of the importance of systematic approaches in mathematics.

The Problem

When each side of a square is increased by 2 cm, its area becomes 81 square centimeters. What is the length of a side of the original square?

Step-by-Step Solution

Let's start by denoting the length of a side of the original square as ( x ) cm.

When each side is increased by 2 cm, the new side length becomes ( x 2 ) area of the new square is given as 81 square centimeters, so we can set up the equation:[ (x 2)^2 81 ]

Now, let's solve for ( x ):

Take the square root of both sides:[ x 2 sqrt{81} ]Since the square root of 81 is 9, we have:[ x 2 9 ]Subtract 2 from both sides:[ x 9 - 2 ]Therefore, the length of a side of the original square is:[ x 7 , text{cm} ]

Verification

To verify our solution, let's consider the new side length which is 9 cm (i.e., ( 7 2 ) cm) and check the area:

The area of the new square is:[ 9^2 81 , text{sq. cm} ]Comparing this with the original side length of 7 cm, the area is:[ 7^2 49 , text{sq. cm} ]So, the difference in area is:[ 81 - 49 32 , text{sq. cm} ]

However, the problem states that the area is increased by 45 sq. cm. So, we can see that the discrepancy is due to the fact that the problem is differently phrased. We need to solve based on the exact given increase, as shown in the subsequent problems.

Advanced Example: Another Scenario

Now, let's consider a similar problem: if each side of a square is increased by 3 cm, its area is increased by 45 sq. cm. What is the length of a side of the original square?

Let ( x ) be the side length of the original equation for the new area is given as:[ (x 3)^2 x^2 45 ]Expanding and simplifying:[ x^2 6x 9 x^2 45 ]Subtract ( x^2 ) from both sides:[ 6x 9 45 ]Subtract 9 from both sides:[ 6x 36 ]Divide by 6:[ x 6 , text{cm} ]

Thus, the length of the side of the original square is 6 cm. Checking the new side length, we have:

The new side length is:[ 6 3 9 , text{cm} ]The new area is:[ 9^2 81 , text{sq. cm} ]The original area is:[ 6^2 36 , text{sq. cm} ]And the increase in area is:[ 81 - 36 45 , text{sq. cm} ]

Conclusion

In both cases, we have successfully solved the problem by using basic algebra and understanding the relationship between the side length and the area of a square. The key steps include setting up the equation correctly, isolating the variable, and verifying the solution to ensure accuracy.