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resuelve es del tema "Rectas y planos en el espacio. Teorema de tales

To solve the problem, we need to use the concept of parallel planes and the application of the **Tales theorem** (also known as the intercept theorem).

### Problem Breakdown:
- The image shows three planes and the line intersecting them, with different segments marked with numerical values and an algebraic expression.
- We have to determine the value of \( x \) such that the planes are parallel.

### Solution:
1. **Identify the ratios**: According to the intercept theorem, if the planes are parallel, the segments created by the transversal line should maintain the same ratio.

   From the image:
   - The segment on the top plane: \( AD = 10 \) and \( DB = 8 \).
   - The segment on the bottom plane: \( CF = 2x - 3 \) and \( FR = 5 \).

2. **Set up the proportion**:
   \[
   \frac{AD}{DB} = \frac{CF}{FR}
   \]
   Substituting the given values:
   \[
   \frac{10}{8} = \frac{2x - 3}{5}
   \]
   Simplify the fraction on the left side:
   \[
   \frac{5}{4} = \frac{2x - 3}{5}
   \]

3. **Cross-multiply to solve for \( x \)**:
   \[
   5 \times 5 = 4 \times (2x - 3)
   \]
   \[
   25 = 8x - 12
   \]
   Add 12 to both sides:
   \[
   37 = 8x
   \]
   Finally, divide by 8:
   \[
   x = \frac{37}{8} = 4.625
   \]

Thus, the value of \( x \) that ensures the planes are parallel is \( x = 4.625 \).

Do you have any questions or would you like further details?

### Related Questions:
1. What is the general form of the intercept theorem?
2. How do you determine if two lines are parallel in three-dimensional space?
3. What happens to the planes if the line segments are not proportional?
4. How is the intercept theorem applied in real-world scenarios?
5. What are the conditions for three planes to be parallel in three-dimensional space?

### Tip:
When solving problems involving proportions, it's crucial to ensure all terms are simplified before cross-multiplying to avoid calculation errors.

Learn how to solve the problem involving rectas y planos en el espacio with the application of the Tales theorem. This problem focuses on parallel planes and proportion in three-dimensional space. Suitable for high school students.

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