Jonas divided a square plot of land into eight equal rectangular parts, as shown in the figure, giving one part to each of his eight children. To fence her part, Antônia found that its perimeter was $120 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-07.jpg?height=333&width=328&top_left_y=510&top_left_x=944)

a) What is the area of the plot of land that Jonas divided?

b) If we represent the perimeter of one of the smaller plots by $P$, what is the original area of the plot in terms of $P$?
Solution

a) Since the land was initially square, after the division, the length of each rectangle is twice its width. Therefore, the width measures $\frac{120}{6}=20 \mathrm{~m}$ and the length $2 \cdot 20=40 \mathrm{~m}$. Thus, the land before the division had a side length of $80 \mathrm{~m}$ and, consequently, an area of $80^{2}=6,400 \mathrm{~m}^{2}$.

b) Repeating the steps from the previous item, the width of each rectangle measures $\frac{P}{6}$ and the length $\frac{P}{3}$. The side of the square measures $\frac{2 P}{3}$ and, consequently, its area $\frac{4 P^{2}}{9}$.