The perimeter of a square is 48 cm. The area of a rectangle is $4 \text{ cm}^2$ less than the area of the square. If the length of the rectangle is 14 cm, then its perimeter is

### Concept & Comparing Area and Perimeter This problem requires moving from the perimeter of a square to its area, relating that to a rectangle's area, and finally determining the rectangle's perimeter using its given length. $$ \text{Perimeter of Square} = 4a $$ $$ \text{Area of Square} = a^2 $$ $$ \text{Perimeter of Rectangle} = 2(l + b) $$ ### Step-by-Step Solution * Given the perimeter of the square is 48 cm: $4a = 48 \Rightarrow a = 12$ cm. * The area of the square is $a^2 = 12^2 = 144 \text{ cm}^2$. * The area of the rectangle is $4 \text{ cm}^2$ less than the square's area: $144 - 4 = 140 \text{ cm}^2$. * We are given the length of the rectangle $l = 14$ cm. Since $\text{Area} = l \times b$, we have $14 \times b = 140 \Rightarrow b = 10$ cm. * The perimeter of the rectangle is $2(l + b) = 2(14 + 10) = 2(24) = 48$ cm. ### Exam Strategy & Shortcut Notice that once you find the area is 140 and the length is 14, the breadth is immediately obvious as 10. The perimeter formula $2(14+10)$ can be calculated mentally in seconds without needing to write down the final steps. ### Common Pitfall A common mistake is confusing the formulas for area and perimeter, such as taking the square root of the perimeter to find the side of the square. ### Final Answer Therefore, the correct answer is **48 cm**.