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.