To determine the perimeter of the triangle given that $DE$ is parallel to $BC$, let's analyze the geometric properties and apply the information provided.

We are given the lengths of various sides in the triangle:

• $AB = 6 \, \text{cm}$
• $AC = 8 \, \text{cm}$
• $BC = 9 \, \text{cm}$
• $AE = 7.5 \, \text{cm}$
• $EC = 9 \, \text{cm}$

Given that $DE \parallel BC$, the triangles $ADE$ and $ABC$ are similar by the Basic Proportionality Theorem (or Thales' Theorem).

To find the perimeter of the triangle $ADE$, we need to calculate the lengths of $DE$, $AD$, and $AE$.

Let's start by setting up the ratio of similarity:

$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$

Given the lengths of $AE = 7.5 \, \text{cm}$ and $AC = 8 \, \text{cm}$, the ratio is:

$\frac{AE}{AC} = \frac{7.5}{15} = \frac{1}{2}$