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The problem asks to determine the value of \( x \) in the given figure, which is a right triangle with some known lengths. Here's the setup:

- The height of the right triangle is 6 units.
- The base of the right triangle is 12 units.
- The hypotenuse, labeled \( x \), needs to be determined.
- A smaller triangle shares the same angle, which forms a proportionate triangle with the same ratio of sides. The base of this smaller triangle is \( 4\sqrt{3} \).

### Strategy
We can solve this problem using the similarity of triangles. Since both triangles share a common angle, the sides of the smaller triangle will be proportional to the sides of the larger triangle.

The larger triangle has:
- Height: \( 6 \)
- Base: \( 12 \)
- Hypotenuse: \( x \) (unknown)

The smaller triangle has:
- Height: unknown (but can be found from the ratio)
- Base: \( 4\sqrt{3} \)
- Hypotenuse: also proportional to the larger triangle.

#### Step 1: Set up a proportion using similar triangles
The ratios of corresponding sides of similar triangles should be equal:
\[
\frac{\text{height of small triangle}}{\text{height of large triangle}} = \frac{\text{base of small triangle}}{\text{base of large triangle}}
\]
Let the height of the small triangle be \( h \).

Thus:
\[
\frac{h}{6} = \frac{4\sqrt{3}}{12}
\]
Simplifying:
\[
\frac{h}{6} = \frac{\sqrt{3}}{3}
\]
\[
h = 6 \times \frac{\sqrt{3}}{3} = 2\sqrt{3}
\]

#### Step 2: Apply the Pythagorean theorem
Now we use the Pythagorean theorem on the larger triangle to find \( x \):
\[
x = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5}
\]

Thus, \( x = 6\sqrt{5} \).

#### Conclusion:
The value of \( x \) is \( 6\sqrt{5} \), which does not directly match any of the provided options (a-d), so the correct answer is option (e) "Ninguna."

Would you like more details on any step?

### Related Questions:
1. How do you know when triangles are similar?
2. What are other methods to determine the length of a triangle's hypotenuse?
3. How can the Pythagorean theorem be applied to non-right triangles?
4. What is the relationship between side lengths and angles in a right triangle?
5. How does the ratio of sides in similar triangles help solve real-world problems?

### Tip:
When dealing with similar triangles, always use the ratio of corresponding sides to find missing lengths.

Determinar el valor x en la siguiente figura (triángulo con base 12, altura 6, y lados en proporción)

Solve a problem involving a right triangle where x is unknown. Using the similarity of triangles and the Pythagorean theorem, this problem explores the relationships between side lengths and angles. Ideal for students in Grades 9-11.

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