To find the value of $x$ in triangle $ABC$, we need to use the concept of similar triangles. When two triangles are similar, their corresponding sides are proportional.

In the image provided:

• Triangles $DEF$ and $ABC$ are similar.
• $DE$ corresponds to $AB$, $DF$ corresponds to $AC$, and $EF$ corresponds to $BC$.

Given:

• $DE = 12$ cm, $EF = 16$ cm.
• $AB = x$ cm, $BC = 32$ cm.

Using the similarity property, we can write the proportion: $\frac{DE}{AB} = \frac{EF}{BC}$

Substitute the given values: $\frac{12}{x} = \frac{16}{32}$

Let's solve this proportion for $x$.The value of $x$ in triangle $ABC$ is $24$ cm.

Would you like a detailed explanation of how the solution was derived, or do you have any other questions? Here are five related questions to explore further:

1. How can we determine if two triangles are similar?
2. What are the properties of similar triangles?
3. How can the Pythagorean theorem be applied to these triangles?
4. If a third triangle is added and is also similar, how do we find its missing sides?
5. What are some real-life applications of similar triangles?

Tip: When working with similar triangles, remember that their corresponding angles are equal and their corresponding sides are in proportion.