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The problem shows two similar triangles, \(\triangle ABC\) and \(\triangle DEF\), and asks to find the missing side length. We know that for similar triangles, corresponding sides are proportional.

The sides of \(\triangle ABC\) are given as:
- \(AB = 12\)
- \(BC = 30\)
- \(AC = 24\)

The sides of \(\triangle DEF\) are:
- \(DE = 2\)
- \(EF = ?\) (the unknown side we need to find)
- \(DF = 4\)

To find the missing side, we can set up a proportion using corresponding sides. Since \(\triangle ABC\) and \(\triangle DEF\) are similar, the ratio of their corresponding sides is the same. Let's compare sides \(AC\) and \(DF\), which are corresponding:

\[
\frac{AC}{DF} = \frac{24}{4} = 6
\]

Now, let's apply this ratio to find side \(EF\) corresponding to side \(BC\). Set up the proportion:

\[
\frac{BC}{EF} = 6 \quad \text{(since the ratio is 6 for corresponding sides)}
\]

Solving for \(EF\):

\[
\frac{30}{EF} = 6 \quad \Rightarrow \quad EF = \frac{30}{6} = 5
\]

So, the missing side length \(EF = 5\).

Would you like more details or have any further questions?

Here are 5 related questions:
1. What defines two triangles as being similar?
2. How would you solve if one of the sides was a decimal instead of a whole number?
3. Can you apply the same process to find unknown angles in similar triangles?
4. How does knowing the ratio between sides help solve problems in real-world applications?
5. How would the process change if only the angles were given?

**Tip:** Always check that the sides you're comparing are indeed corresponding, as the proportions only hold for corresponding pairs.

ΔABC and ΔDEF are similar triangles. Given the sides AB = 12, BC = 30, and AC = 24 for triangle ABC, and DE = 2, DF = 4 for triangle DEF, find the missing side length EF.

Learn how to find the missing side length EF in ΔDEF when ΔABC and ΔDEF are similar triangles. This solution uses the proportionality of corresponding sides and is suitable for students in Grades 8-10.

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