https://r2.erweima.ai/mathbot/image/17259408200965g58y1ru.jpg

The problem provided in the image asks to calculate the "media proporcional" or the geometric mean in the given proportions. For each equation, you need to find the value of \( x \) such that the proportions remain equal. 

To solve these proportions, you can use cross-multiplication to find \( x \). Let’s solve each one:

### a. \(\frac{3}{x} = \frac{x}{7}\)

Cross-multiply:
\[
3 \times 7 = x \times x
\]
\[
21 = x^2
\]
Solving for \( x \):
\[
x = \sqrt{21}
\]

### b. \(\frac{2}{x} = \frac{x}{18}\)

Cross-multiply:
\[
2 \times 18 = x \times x
\]
\[
36 = x^2
\]
Solving for \( x \):
\[
x = \sqrt{36} = 6
\]

### c. \(\frac{4}{x} = \frac{x}{16}\)

Cross-multiply:
\[
4 \times 16 = x \times x
\]
\[
64 = x^2
\]
Solving for \( x \):
\[
x = \sqrt{64} = 8
\]

### d. \(\frac{0}{x} = \frac{x}{10}\)

This proportion has no solution because the numerator is 0, which makes the left-hand side always 0. However, the right-hand side cannot be zero unless \( x = 0 \), but dividing by zero would make the expression undefined. Hence, **no solution**.

### e. \(\frac{5}{x} = \frac{x}{125}\)

Cross-multiply:
\[
5 \times 125 = x \times x
\]
\[
625 = x^2
\]
Solving for \( x \):
\[
x = \sqrt{625} = 25
\]

### f. \(\frac{9}{x} = \frac{x}{8}\)

Cross-multiply:
\[
9 \times 8 = x \times x
\]
\[
72 = x^2
\]
Solving for \( x \):
\[
x = \sqrt{72} \approx 8.49
\]

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Let me know if you need further clarification or more details on any step!

Here are 5 related questions:
1. How do we solve proportions involving square roots?
2. What happens if both sides of the proportion involve variables?
3. Can proportions have negative values for \( x \)? If so, when?
4. How does cross-multiplication work in more complex algebraic proportions?
5. What are the conditions for a proportion to have no solution?

**Tip**: Always check if both sides of the proportion have valid denominators (i.e., they are not zero) before proceeding with calculations.

En las siguientes proporciones, los medios, que están marcados con x tienen el mismo valor, y se denominan la media proporcional. Calcúlalos.

Learn how to calculate the geometric mean (media proporcional) using cross-multiplication in proportion problems. This exercise is suitable for students in grades 8-10, covering important concepts like proportions and the application of the geometric mean.

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