Since the triangles $\triangle ABC$ and $\triangle DEF$ are similar (denoted by $\triangle ABC \sim \triangle DEF$), their corresponding side lengths are proportional. This means that we can set up a proportion between the corresponding sides.

From the diagram:

• Side $AB$ in $\triangle ABC$ corresponds to side $DE$ in $\triangle DEF$.
• Side $AC$ in $\triangle ABC$ corresponds to side $DF$ in $\triangle DEF$.
• Side $BC$ in $\triangle ABC$ corresponds to side $EF$ in $\triangle DEF$.

The given side lengths are:

• $AB = x$
• $BC = 28$
• $AC = 25$
• $DE = 25$
• $EF = 20$

We can set up the following proportion for corresponding sides:

$\frac{AB}{DE} = \frac{BC}{EF}$

Substitute the known values:

$\frac{x}{25} = \frac{28}{20}$

Now, solve for $x$ by cross-multiplying:

$x \cdot 20 = 28 \cdot 25$

$20x = 700$

$x = \frac{700}{20} = 35$

So, the value of $x$ is:

$\boxed{35}$

Would you like further details or have any other questions?

Related Questions

1. How can we determine if two triangles are similar?
2. How do the properties of similar triangles help in solving real-world problems?
3. What other geometric figures have similar properties like triangles?
4. How is proportionality used in finding missing values in polygons?
5. What are some other methods to prove triangle similarity?

Tip

In problems involving similar triangles, always look for corresponding side pairs to set up a proportion for solving unknown values.