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The exercise is about a trapezoid \( ABCD \) where \( BC \parallel AD \). Points \( E \) and \( F \) are the midpoints of \( BC \) and \( AD \), respectively. We are given the vector equation for \( \overrightarrow{EF} \):

\[
\overrightarrow{EF} = m \overrightarrow{DB} + n \overrightarrow{CA}
\]

We are tasked with finding \( m^2 + n^2 \).

### Approach:

1. **Geometry of the Trapezoid:**
   - Since \( BC \parallel AD \), this implies symmetry in some aspects of the vector decomposition.
   - The points \( E \) and \( F \) being midpoints will allow us to use the midpoint theorem, which relates the line segment joining the midpoints to the parallel sides.

2. **Vector Decomposition:**
   - We need to express \( \overrightarrow{EF} \) in terms of vectors \( \overrightarrow{DB} \) and \( \overrightarrow{CA} \), which involve the parameters \( m \) and \( n \).

3. **Solving for \( m \) and \( n \):**
   - We can apply properties of trapezoids and midpoints to write expressions for \( \overrightarrow{EF} \), then compare coefficients to find \( m \) and \( n \).

Once we solve for \( m \) and \( n \), we can compute \( m^2 + n^2 \).

Let me calculate that for you.The value of \( m^2 + n^2 \) is \( 0.5 \).

This result follows from the properties of midpoints in trapezoids, which typically leads to equal contributions from the vectors \( \overrightarrow{DB} \) and \( \overrightarrow{CA} \) when using a symmetrical setup.

Do you want more details on the process or clarification on how these values are derived?

Here are 5 related questions to deepen your understanding:
1. What is the midpoint theorem, and how is it applied in trapezoids?
2. How do parallel lines in trapezoids affect vector decomposition?
3. What other properties can be derived from midpoints in quadrilaterals?
4. Can you extend this analysis to non-parallel quadrilaterals?
5. How does the orientation of the trapezoid influence the vector equation?

**Tip:** In geometry, the symmetry of shapes like trapezoids often simplifies vector equations, especially when dealing with midpoints.

Sobre un trapecio ABCD (BC // AD) se toman los E y F, puntos medios de BC y AD respectivamente. Se sabe que: EF = mDB + nCA. Hallar m² + n².

Learn how to solve for m² + n² in a trapezoid where BC is parallel to AD, and EF is the line joining midpoints E and F. This problem involves vector algebra and the midpoint theorem, suitable for students in Grades 10-12.

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