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This problem involves finding the length of the tunnel using trigonometry, as the diagram forms a triangle with known distances and angles.

### Known Values:
- \( \text{Distance from the observer to the left side of the tunnel} = 239 \, \text{m} \)
- \( \text{Distance from the observer to the right side of the tunnel} = 135 \, \text{m} \)
- \( \text{Angle between the two lines of sight} = 77^\circ \)

### Steps to Solve:
1. **Label the Triangle:**
   - Let the observer be at point \( O \).
   - Let the left side of the tunnel be point \( A \) (239 m away).
   - Let the right side of the tunnel be point \( B \) (135 m away).
   - The angle \( \angle AOB = 77^\circ \).
   - We are tasked with finding the length of the tunnel, which is the distance between points \( A \) and \( B \) (denoted as \( AB \)).

2. **Apply the Law of Cosines:**
   The Law of Cosines is useful in this case because we know two sides and the included angle. The formula is:
   \[
   AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos(\angle AOB)
   \]
   Substituting the known values:
   \[
   AB^2 = 239^2 + 135^2 - 2 \cdot 239 \cdot 135 \cdot \cos(77^\circ)
   \]
   First, calculate each part:
   - \( 239^2 = 57121 \)
   - \( 135^2 = 18225 \)
   - \( \cos(77^\circ) \approx 0.22495 \)
   - \( 2 \cdot 239 \cdot 135 \cdot 0.22495 = 14501.05 \)

   Now plug these into the formula:
   \[
   AB^2 = 57121 + 18225 - 14501.05
   \]
   \[
   AB^2 = 60844.95
   \]
   Taking the square root to find \( AB \):
   \[
   AB = \sqrt{60844.95} \approx 246.7 \, \text{m}
   \]

Thus, the length of the tunnel is approximately **246.7 meters**.

Would you like a detailed explanation of any step? Here are some related questions:

1. How do we calculate angles using the Law of Cosines?
2. What is the significance of rounding in trigonometric calculations?
3. How does changing the angle affect the tunnel length in this case?
4. Could this problem be solved using the Law of Sines instead?
5. How does the angle measurement method impact practical surveying?

**Tip**: Always use a calculator with accurate trigonometric functions when dealing with angles to minimize rounding errors.

Boris wants to know the length of a tunnel built through a mountain. Given the distances of 239 meters and 135 meters from the observer to the left and right sides of the tunnel, respectively, and an angle of 77° between these two lines of sight, calculate the length of the tunnel.

Learn how to find the length of a tunnel using trigonometry and the Law of Cosines. The problem involves distances of 239 meters and 135 meters from an observer and an angle of 77° between them. A detailed step-by-step solution is provided, ideal for students in Grades 9-12.

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