https://r2.erweima.ai/mathbot/image/17260257998616saifw1u.jpg

To solve the problem, let's break it down step-by-step:

### Problem Analysis:

Given:
- Trapezoid \( ABCD \) where \( BC \parallel AD \).
- \( E \) and \( F \) are midpoints of \( BC \) and \( AD \), respectively.
- The vector \( \overrightarrow{EF} = m\overrightarrow{DB} + n\overrightarrow{CA} \).
- We need to find \( m^2 + n^2 \).

### Step-by-Step Solution:

1. **Understand the Geometry of the Problem:**
   - \( ABCD \) is a trapezoid with \( BC \parallel AD \).
   - \( E \) and \( F \) are the midpoints of \( BC \) and \( AD \), respectively.

2. **Apply the Midpoint Theorem:**
   - Since \( E \) and \( F \) are midpoints, the line segment \( EF \) is the midsegment of the trapezoid.
   - The midsegment of a trapezoid is parallel to the bases and its length is half the sum of the lengths of the bases. Thus, \( EF \parallel BC \parallel AD \).

3. **Express \( \overrightarrow{EF} \) in Terms of Vectors:**
   - Let's denote \( \overrightarrow{BC} = \mathbf{b} \) and \( \overrightarrow{AD} = \mathbf{a} \).
   - Since \( EF \) is the midsegment, \( \overrightarrow{EF} = \frac{1}{2}(\overrightarrow{AD} + \overrightarrow{BC}) = \frac{1}{2}(\mathbf{a} + \mathbf{b}) \).

4. **Express \( \overrightarrow{DB} \) and \( \overrightarrow{CA} \) in Terms of \( \mathbf{a} \) and \( \mathbf{b} \):**
   - \( \overrightarrow{DB} = \overrightarrow{DA} + \overrightarrow{AB} = -\mathbf{a} + \overrightarrow{AB} \).
   - \( \overrightarrow{CA} = \overrightarrow{CB} + \overrightarrow{BA} = \mathbf{b} - \overrightarrow{AB} \).

5. **Find \( m \) and \( n \):**
   - Since \( \overrightarrow{EF} = m\overrightarrow{DB} + n\overrightarrow{CA} \), substitute the expressions we derived:
     \[
     \frac{1}{2}(\mathbf{a} + \mathbf{b}) = m(-\mathbf{a} + \overrightarrow{AB}) + n(\mathbf{b} - \overrightarrow{AB})
     \]
   - We equate the coefficients of \( \mathbf{a} \), \( \mathbf{b} \), and \( \overrightarrow{AB} \) from both sides to solve for \( m \) and \( n \).

6. **Calculate \( m^2 + n^2 \):**
   - By solving the system of linear equations obtained from the coefficients, we find \( m \) and \( n \). 
   - Once we have the values of \( m \) and \( n \), we calculate \( m^2 + n^2 \).

Let me calculate the values of \( m \) and \( n \).The values obtained for \( m \) and \( n \) are:

\[
m = -\frac{1}{2}, \quad n = \frac{1}{2}
\]

Thus, the value of \( m^2 + n^2 \) is:

\[
m^2 + n^2 = \left(-\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}
\]

### Conclusion:
\[
m^2 + n^2 = \frac{1}{2}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the midsegment of a trapezoid?
2. What are the properties of vectors in geometric proofs?
3. How can we use vectors to solve problems involving parallelograms?
4. How can the midpoint theorem be used in different geometric shapes?
5. What are some examples of vector applications in physics?

### Tip:
When solving problems with vectors, it's helpful to break them down into components and relate them to known points or directions.

Sobre un trapecio ABCD (BC // AD) se toman los E y F, puntos medios de BC y AD respectivamente. Si se sabe que: EF = mDB + nCA. Hallar m^2 + n^2.

In this problem, we analyze a trapezoid ABCD with midpoints E and F on sides BC and AD, respectively. Using the midpoint theorem and vector properties, we solve for m^2 + n^2 from the equation EF = mDB + nCA. This problem covers vector geometry concepts and is ideal for students in Grades 10-12.

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