该图形由2个相同的三角形和2个相同的正方形组成。该图形的周长为 $(4q + 36)$ cm。正方形的周长为 36 cm。

1. Find the side length of the square

$\text{Side of square} = 36 \div 4 = 9\text{ cm}$

2. Find the sum of the exposed sides of the two triangles

The perimeter of the figure consists of 4 square sides ($AH, HG, GF, FE$) and 4 triangle sides ($AB, BC, CD, DE$).
$\text{Sum of 4 square sides} = 4 \times 9 = 36\text{ cm}$
$\text{Sum of 4 triangle sides} = (4q + 36) - 36 = 4q\text{ cm}$

3. Find the sum of the two exposed sides of one triangle

Since the two triangles are identical, the sum of the two exposed sides of triangle ABC ($AB + BC$) is:
$4q \div 2 = 2q\text{ cm}$

4. Calculate the perimeter of triangle ABC

The perimeter of triangle ABC includes the two exposed sides ($AB + BC$) and the base ($AC$), which is a side of the square.
$\text{Perimeter of } \triangle ABC = 2q + 9\text{ cm}$

1. Equate the expression for the perimeter to the given value

$4q + 36 = 48$

2. Solve the equation for $q$

$4q = 48 - 36$
$4q = 12$
$q = 3$