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The problem you shared is asking for the area of quadrilateral \(BDEC\), given certain details about the triangle \(\triangle ABC\) and \(\triangle ADE\).

From the diagram and the provided information, we know the following:
- \( BD = 5 \)
- \( AE = 3 \)
- \( CE = 6 \)
- The area of triangle \(ADE\) is given as 5 square units.

### Steps to solve for the area of quadrilateral \(BDEC\):

1. **Use the area ratio in similar triangles:**
   - The triangles \(\triangle ABC\) and \(\triangle ADE\) are likely similar because they share the same angle at \(A\) (based on the diagram's layout).
   - The sides of the triangle provide a ratio between the corresponding sides. For instance, the ratio of side \(AE\) to \(AC\) can be used to find the area ratio.

2. **Find the full area of triangle \(ABC\):**
   - Once we know the ratio between \(\triangle ADE\) and \(\triangle ABC\), we can use the area of \(\triangle ADE\) to find the area of \(\triangle ABC\).
   
3. **Subtract the area of \(\triangle ADE\) from \(\triangle ABC\):**
   - After finding the area of \(\triangle ABC\), subtract the area of \(\triangle ADE\) from it to find the area of quadrilateral \(BDEC\).

Let me calculate these steps for you.The area of quadrilateral \(BDEC\) is 40 square units. 

However, this result doesn't match any of the multiple-choice answers (10, 15, 20, 25), which suggests there might be additional constraints or assumptions in the problem that need reconsideration. Would you like more details on any specific part of the solution?

Here are 5 related questions to think about:
1. How do the properties of similar triangles help in finding areas?
2. Can we use another method, such as the Pythagorean theorem, to verify distances in this diagram?
3. How does the ratio of sides affect the area of similar triangles?
4. What if triangle \(ADE\) is not similar to triangle \(ABC\)? How would we approach this differently?
5. Could the missing step involve using a different part of the triangle's geometry?

**Tip:** Always double-check the given dimensions and diagram for any hidden relationships or symmetries that might simplify the calculations.

In triangle ABC, AD is the height, BD = 5, AE = 3, and CE = 6. Given that the area of triangle ADE is 5 square units, what is the area of quadrilateral BDEC?

Learn how to calculate the area of quadrilateral BDEC using properties of similar triangles and the given dimensions of triangle ADE and triangle ABC. Suitable for students in Grades 9-10, this problem focuses on geometric concepts such as area calculation and triangle similarity.

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