Two sides of a square are represented by \( (x+9) \) and \( (6 x-1) \). What is the perimeter of the square? Show all work. POSSIBLE
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To find the perimeter of the square, we first need to establish that both sides are equal because all sides of a square are of equal length. Thus, we set the two expressions equal to each other: \[ x + 9 = 6x - 1 \] Next, we'll solve for \( x \). To do this, we first subtract \( x \) from both sides: \[ 9 = 5x - 1 \] Now, we add 1 to both sides to isolate the term with \( x \): \[ 10 = 5x \] Next, we divide both sides by 5: \[ x = 2 \] Now that we’ve determined the value of \( x \), we can substitute it back into either expression for the sides. We can use \( (x + 9) \): \[ \text{Side} = x + 9 = 2 + 9 = 11 \] Since it's a square, all sides are equal to 11. The formula for the perimeter \( P \) of a square is given by: \[ P = 4 \times \text{Side} \] Substituting the value we found: \[ P = 4 \times 11 = 44 \] Thus, the perimeter of the square is \( \boxed{44} \). The history behind the development of geometric shapes like squares traces back to ancient civilizations, where mathematicians explored the relationship between length, area, and perimeter. The square's distinct properties were foundational in the evolution of geometry. In the real world, understanding squares and their properties is vital in various fields, from architecture to design. Architects use the principles of squares when planning the layout of buildings, ensuring balanced and aesthetically pleasing structures while maximizing space efficiency.
