B is 844 metres due south of A. C is due west of A. The bearing of C from B is 295 degrees. Find the distance BC, to the nearest metre.**#**

To solve for the distance BC given that B is 844 meters due south of A, C is due west of A, and the bearing of C from B is 295 degrees, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Bearings:**
   - Bearings are measured clockwise from the north direction.
   - A bearing of 295 degrees from B to C means that the angle from the north line at B to the line BC is 295 degrees.

2. **Convert Bearing to Angle:**
   - Bearing of 295 degrees from B means \( 360^\circ - 295^\circ = 65^\circ \) west of the north line from B.

3. **Calculate the Angle with South Line:**
   - Since B is due south of A, the line from A to B is a vertical line pointing north-south.
   - The angle from the south line at B to the line BC is \( 180^\circ - 65^\circ = 115^\circ \).

4. **Set Up the Problem Using Trigonometry:**
   - Let \( \angle ABC \) be 115 degrees (the angle between line AB and line BC).
   - AB = 844 meters (distance from A to B).

5. **Use the Law of Sines:**
   - In triangle ABC:
     \[
     \frac{BC}{\sin(115^\circ)} = \frac{AB}{\sin(25^\circ)}
     \]
     \[
     BC = \frac{AB \cdot \sin(115^\circ)}{\sin(25^\circ)}
     \]

6. **Calculate the Values:**
   - \(\sin(115^\circ) = \sin(65^\circ) \approx 0.9063\)
   - \(\sin(25^\circ) \approx 0.4226\)

     Substituting these into the equation:
     \[
     BC = \frac{844 \cdot 0.9063}{0.4226}
     \]
     \[
     BC \approx \frac{764.9172}{0.4226} \approx 1809.55 \text{ meters}
     \]

7. **Rounding to the Nearest Meter:**
   - The distance BC is approximately 1810 meters.

Thus, the distance \( BC \) to the nearest meter is **1810 meters**.

Would you like any further details or explanations?

### Further Questions:
1. What is the Law of Sines, and how is it used in trigonometry?
2. How do you convert a bearing to a standard angle for calculations?
3. What are some practical applications of solving distance problems using bearings?
4. How does the Pythagorean theorem differ from the Law of Sines?
5. What steps are essential in verifying the accuracy of such trigonometric calculations?
6. How do you handle non-right-angled triangles in trigonometric problems?
7. What tools can assist in solving complex bearing and distance problems?
8. How can understanding bearings and distances aid in navigation and mapping?

### Tip:
Always draw a detailed diagram of the problem when dealing with bearings and distances. It helps visualize angles and relationships between different points, making the application of trigonometric laws more intuitive.

Learn how to calculate the distance BC using bearings and coordinates. This problem involves trigonometric calculations, the Law of Sines, and practical applications of bearings in navigation. Suitable for advanced high school students.

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