The figure shows nine identical rectangles fitted together to make a large rectangle whose area is 72 cm².
Find the perimeter of each small rectangle.

Let the dimensions of each small rectangle be:

$$\begin{aligned} \text{length} &= l \\ \text{breadth} &= b \end{aligned}$$

From the figure: Top row has 4 rectangles placed vertically

Width of large rectangle = $4b$

Bottom row has 5 rectangles placed horizontally

Width of large rectangle = $5l$

Since both widths are equal, $4b = 5l$ ...(i)

The height of the large rectangle is $l + b$.

So, the area of the large rectangle is $b(l + b) = 72$.

Using $4b = 5l$,

$$5l(l + b) = 72$$

From (i), $b = \frac{5l}{4}$

Substitute: $5l\left(l + \frac{5l}{4}\right) = 72$

$$\frac{45l^2}{4} = 72$$

$$45l^2 = 288$$

$$l^2 = \frac{288}{45} = \frac{32}{5}$$

$$l = \frac{4\sqrt{10}}{5}$$

Then, $b = \frac{5}{4} \times \frac{4\sqrt{10}}{5} = \sqrt{10}$

Perimeter of one small rectangle:

$$\begin{aligned} P &= 2(l + b) \\ &= 2\left(\frac{4\sqrt{10}}{5} + \sqrt{10}\right) \\ &= 2\left(\frac{9\sqrt{10}}{5}\right) \\ &= \frac{18\sqrt{10}}{5} \text{ cm} \end{aligned}$$