The problem given states that $\triangle ABC \parallel \triangle DEC$, meaning the triangles are similar. In similar triangles, corresponding sides are proportional.

We are given the following information:

For $\triangle ABC$:

• $AB = x$
• $BC = 2 \, \text{cm}$
• $AC = 3 \, \text{cm}$

For $\triangle DEC$:

• $DE = 12 \, \text{cm}$
• $EC = 9 \, \text{cm}$
• $DC = 6 \, \text{cm}$

Using the property of similar triangles, the ratio of the corresponding sides will be equal:

$\frac{AB}{DE} = \frac{BC}{EC} = \frac{AC}{DC}$

Let's set up these proportions:

$\frac{x}{12} = \frac{2}{9} = \frac{3}{6}$

First, simplify the ratio $\frac{3}{6}$:

$\frac{3}{6} = \frac{1}{2}$

Thus, the proportion becomes:

$\frac{x}{12} = \frac{2}{9} = \frac{1}{2}$

Now, solve for $x$ using $\frac{x}{12} = \frac{1}{2}$:

$\frac{x}{12} = \frac{1}{2}$

Cross-multiply:

$x \times 2 = 12 \times 1$

$2x = 12$

Divide by 2:

$x = 6$

Therefore, the size of $x$ is 6 cm.

Would you like more details or have any questions? Here are five related questions to deepen your understanding:

1. How do you know which sides correspond in similar triangles?
2. What other ways can you prove that two triangles are similar?
3. Can you apply the same proportional reasoning if only one pair of sides is given?
4. How do you calculate the height of a triangle given its sides and base?
5. What would happen if the triangles were not similar?

Tip: Always check if two triangles are similar by verifying their angles or corresponding sides. This helps you set up the correct proportions!