Hello again, polakio92!

c) A rectangle is cut to form a square.

The perimeter of the rectangle is 54 cm, and the area of the cut-off piece is 36 cm².

What are the dimensions of the original rectangle? Code:

x y
* - - - - - * - - - *
| | |
| | |
x | | | x
| | |
| | |
* - - - - - * - - - *
x y

The perimeter of the rectangle is 54: .$\displaystyle 4x + 2y \:=\:54\quad\Rightarrow\quad y \:=\:27 - 2x\;\;{\color{blue}[1]}$

The area of the cut-off piece is 36 cm²: .$\displaystyle xy \:=\:36\;\;{\color{blue}[2]}$

Substitute [1] into [2]: .$\displaystyle x(27-2x) \:=\:36\quad\Rightarrow\quad 2x^2 - 27x + 36 \:=\:0$

. . which factors: .$\displaystyle (x-12)(2x-3)\:=\:0$

. . and has roots: .$\displaystyle x \;=\;12,\:\frac{3}{2}$

and has y-values: .$\displaystyle y \;=\;3,\:24$

There are **two** solutions: .$\displaystyle (x,y) \;=\;(12,\,3)\text{ and }(1.5,\:24)$

The original rectangle was: . $\displaystyle \boxed{12 \times 15}\;\text{ or }\;\boxed{1.5 \times 25.5}$

Code:

12 3
* - - - - - * - - - *
| | |
| | |
12 | | | 12
| | |
| | |
* - - - - - * - - - *
12 3
1½ 24
* - * - - - - - - - - - - - - *
1½ | | | 1½
* - * - - - - - - - - - - - - *
1½ 24

Both of them satisfy the problem!