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Determine la distancia AB con los datos que se muestran en la figura
Nota: Ingrese la respuesta con un decimal de precisión 

To determine the distance \(AB\) based on the provided diagram, we can apply the Law of Cosines. The diagram shows two triangles, with the angles and sides given. Let's break down the problem into steps:

### Step 1: Identify the triangles and the given data
- Triangle at point \(A\) with angles \(54^\circ\), \(60^\circ\), and \(66^\circ\).
- Triangle at point \(B\) with angles \(54^\circ\), \(60^\circ\), and \(66^\circ\).
- Side adjacent to angle \(66^\circ\) in the triangle at \(A\) is 16.35.
- Side adjacent to angle \(66^\circ\) in the triangle at \(B\) is 19.67.

### Step 2: Applying the Law of Cosines
For each triangle, we can use the Law of Cosines to determine the distance from the intersection to \(A\) and \(B\), and then add them up to find \(AB\).

**For Triangle at A:**

Let's denote the distance from the intersection point (point \(X\)) to \(A\) as \(AX\), and similarly, for \(BX\):

\[ AX^2 = 16.35^2 + d^2 - 2(16.35)(d)\cos(60^\circ) \]
Where \(d\) is the unknown distance \(AX\).

**For Triangle at B:**

Similarly:

\[ BX^2 = 19.67^2 + d^2 - 2(19.67)(d)\cos(60^\circ) \]

### Step 3: Calculation

We need to calculate the distance \(AB = AX + BX\). The given angles and side lengths provide the necessary data to compute these lengths.

### Final Answer:

Let me calculate this in detail.The distance \( AB \) is approximately \( 36.0 \) units when rounded to one decimal place.

Would you like any further details or have any questions?

Here are 5 related questions to expand on the topic:

1. How does the Law of Cosines differ from the Law of Sines, and when is each used?
2. Can the Law of Cosines be applied to any type of triangle, or are there specific conditions?
3. How would the calculation change if the angle between the sides was not \(60^\circ\)?
4. What are the implications if the diagram represents a real-world scenario like a bridge?
5. How can the Law of Cosines be applied to find the angles if the sides are known?

**Tip:** Always double-check the angle measurements and side lengths in a problem like this, as even a small mistake can significantly impact the result.

Learn how to calculate the distance AB using the Law of Cosines with a step-by-step solution. This problem involves trigonometric principles and geometric calculations. Suitable for high school students.

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