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To solve this problem, we will use the information provided to determine the length of side \( AB \) in triangle \( ABC \), where \( \angle ABC = 90^\circ \).

### Information Given:
1. Triangle \( ABC \) is a right triangle (\( \angle ABC = 90^\circ \)).
2. Line \( AC \) has an azimuth of \( 71^\circ \) from point \( A \) and length of \( 141.1 \, \text{m} \).
3. Line \( CB \) has a bearing of \( S 29^\circ W \) (from point \( C \)).

### Step 1: Analyze the Azimuth and Bearing
- The azimuth is measured clockwise from the North. Hence, \( AC \) makes an angle of \( 71^\circ \) with the North direction.
- The bearing \( S 29^\circ W \) implies that line \( CB \) forms a \( 29^\circ \) angle with the South-West quadrant.

To find the lengths of the sides \( AB \) and \( BC \), we can use trigonometric relationships, including sine and cosine laws, or vector decomposition. 

### Step 2: Apply the Law of Cosines or Trigonometry to Find AB
We will solve for \( AB \) using the angles and distances given.

I'll calculate the result now.The length of side \( AB \) is approximately \( 123.41 \, \text{m} \), which rounds to \( 120 \, \text{m} \) to two significant digits.

Do you want further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How is azimuth different from a bearing?
2. Why do we subtract \( 90^\circ + 29^\circ \) to find angle \( ACB \)?
3. What are the main applications of azimuths and bearings in navigation?
4. How does the Law of Sines apply to right triangles?
5. How would the problem change if \( \angle ABC \) were not \( 90^\circ \)?

**Tip:** Always double-check your angle measurements when working with bearings and azimuths to avoid confusion between compass directions and mathematical angles.

For the triangle ABC, where angle ABC is 90°, find length AB. Line AC has an azimuth of 71° (origin point is A) and length of 141.1 m. Line CB has a bearing of S 29° W (origin point is C). Provide the answer in metres, round it to 2 significant digits, and enter the unit.

Explore how to determine the length of side AB in a right triangle using given bearings and azimuths for lines AC and CB, with detailed steps and trigonometric applications. Suitable for high school students.

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