https://file.aiquickdraw.com/mathbot/file/1726758213315hcqlkmmk.jpg

The diagram from the uploaded image indicates two similar triangles, \( \triangle ABC \) and \( \triangle DBE \), with corresponding angles and sides.

Given:
- \( AC = 27 \)
- \( BC = 15 \)
- \( DE = x \)
- \( BE = 10 \)

The triangles are similar because they have two angles in common, \( \angle ACB = \angle DEB = 130^\circ \) and \( \angle ABC = \angle DBE \).

We use the similarity ratio to find \( DE \):
\[
\frac{AC}{BC} = \frac{DB}{BE}
\]
Substitute the known values:
\[
\frac{27}{15} = \frac{DB}{10}
\]
Solving for \( DB \):
\[
DB = \frac{27}{15} \times 10 = 18
\]
Thus, \( DE = 18 \).

So, the length of \( DE \) is \( 18 \). The correct answer is \( C. 18 \).

Would you like more details or do you have any other questions? Here are some potential follow-up queries:
1. How can you prove that the triangles are similar beyond using angles?
2. What would happen if one of the angles changed slightly?
3. Can you apply this method to find the length of \( AC \) if \( DE \) and other dimensions were known?
4. How does the similarity concept apply in real-world scenarios?
5. What are other mathematical properties that triangles share beyond similarity?

**Tip:** Always double-check whether the triangles share a common angle or side when applying similarity rules to ensure the calculation is valid.

Learn how to solve for the length of DE using the similarity of triangles, given that AC = 27, BC = 15, BE = 10, and corresponding angles. This problem applies the AA similarity theorem and uses proportions to find the missing side. Suitable for students in Grades 9-10.

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