Question

stacey has a square piece of cloth. she cuts 3 inches off of the length of the square and 3 inches off of the width. the area of the smaller square is \\(\frac{1}{4}\\) the area of the original square. what was the side length of the original square?
\\(a = s^2\\)
\\(\bigcirc\\) 1 in
\\(\bigcirc\\) 2 in
\\(\bigcirc\\) 6 in
\\(\bigcirc\\) 12 in

Explanation:

Step1: Define variables

Let the side length of the original square be \( s \) inches. The area of the original square is \( A_{original}=s^2 \). After cutting 3 inches from length and width, the side length of the smaller square is \( s - 3 \) inches, so its area is \( A_{smaller}=(s - 3)^2 \).

Step2: Set up equation

We know that the area of the smaller square is \( \frac{1}{4} \) of the original square's area. So, \( (s - 3)^2=\frac{1}{4}s^2 \).

Step3: Solve the equation

Take square roots on both sides (considering positive roots since side length is positive): \( s - 3=\frac{1}{2}s \) (we take the positive root because \( s-3 \) and \( s \) are positive lengths).
Subtract \( \frac{1}{2}s \) from both sides: \( s-\frac{1}{2}s - 3 = 0 \)
Simplify: \( \frac{1}{2}s-3 = 0 \)
Add 3 to both sides: \( \frac{1}{2}s=3 \)
Multiply both sides by 2: \( s = 6 \). We can check: original area \( 6^2 = 36 \), smaller side \( 6 - 3 = 3 \), smaller area \( 3^2 = 9 \), and \( 9=\frac{1}{4}\times36 \), which holds.

Answer:

6 in (corresponding to the option "6 in")