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The problem shows two similar right triangles \( \triangle ABC \) and \( \triangle DEF \). We are tasked with finding the height of \( \triangle DEF \), rounding to the nearest tenth. 

Since the triangles are similar, the corresponding sides are proportional. The height of triangle \( \triangle ABC \) is 3 m (between point \( A \) and point \( C \)) and its base is 2 m (from \( A \) to \( B \)). The base of triangle \( \triangle DEF \) is 5 m (from \( D \) to \( E \)).

We can set up a proportion based on the corresponding sides:

\[
\frac{\text{height of } \triangle ABC}{\text{base of } \triangle ABC} = \frac{\text{height of } \triangle DEF}{\text{base of } \triangle DEF}
\]

\[
\frac{3}{2} = \frac{\text{height of } \triangle DEF}{5}
\]

Now, solve for the height of \( \triangle DEF \):

\[
\text{height of } \triangle DEF = \frac{3}{2} \times 5 = 7.5 \, \text{m}
\]

Thus, the height of triangle \( DEF \) is approximately **7.5 meters**.

Would you like more details or have any questions?

Here are some follow-up questions to expand on the concept:

1. How do we know that triangles \( ABC \) and \( DEF \) are similar?
2. How would the problem change if the triangles were not similar?
3. Can you explain why the ratio of corresponding sides must be equal for similar triangles?
4. How could you apply this concept of similarity to other shapes, such as trapezoids or polygons?
5. What are other methods to find the height of a triangle, besides using proportions?

**Tip:** In any problem involving similar triangles, always ensure you correctly identify the corresponding sides before setting up the proportion.

Triangles ABC and DEF are similar triangles. Solve. Round to the nearest tenth. Find the height of triangle DEF.

This problem involves using similar triangles and proportions to find the height of triangle DEF. Given that the triangles are similar, you can set up a proportion based on the corresponding sides to solve for the unknown height. The problem is suitable for students in grades 7-10 and provides an application of the similar triangles theorem.

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