The bearing of a point A from P is 310o and the bearing of another point B from P is 292o. Both A and B are due north of a point C which is 5600 m due west of P.

Find the distance between A and B.

To solve the problem, we'll break it down step by step.

### Step 1: Understand the Bearing Information
- The bearing of a point is the direction measured in degrees clockwise from the north.
- The bearing of point A from point P is \(310^\circ\).
- The bearing of point B from point P is \(292^\circ\).
  
Since bearings are measured clockwise from the north:
- \(310^\circ\) means A is slightly west of north.
- \(292^\circ\) means B is even further west of north.

### Step 2: Visualize and Set Up the Triangle
We are given that both points A and B are due north of a point C, which is 5600 meters due west of P. This setup implies that the distance between A and B is simply the horizontal (west-east) distance between them, as both points lie on the same vertical line (due north).

Given that bearings are measured clockwise from the north:
- The angle \(\theta_1\) from the north to PA (from P to A) is \(360^\circ - 310^\circ = 50^\circ\).
- The angle \(\theta_2\) from the north to PB (from P to B) is \(360^\circ - 292^\circ = 68^\circ\).

### Step 3: Calculate the Horizontal Distances to Points A and B
Using trigonometry, we can calculate the horizontal distances (westward components) to points A and B.

For point A:
\[
x_A = \text{Distance from P to C} \times \tan(\theta_1) = 5600 \times \tan(50^\circ)
\]

For point B:
\[
x_B = \text{Distance from P to C} \times \tan(\theta_2) = 5600 \times \tan(68^\circ)
\]

### Step 4: Calculate the Distance Between A and B
The distance between A and B, \(d_{AB}\), is the difference between the horizontal distances to A and B:
\[
d_{AB} = |x_B - x_A|
\]

Let’s calculate it.

\[
x_A = 5600 \times \tan(50^\circ) \approx 5600 \times 1.1918 \approx 6674.08 \text{ meters}
\]
\[
x_B = 5600 \times \tan(68^\circ) \approx 5600 \times 2.4751 \approx 13860.56 \text{ meters}
\]

Thus, the distance between A and B is:
\[
d_{AB} = |13860.56 - 6674.08| \approx 7186.48 \text{ meters}
\]

### Final Answer
The distance between points A and B is approximately **7186.48 meters**.

Would you like further details on any part of the solution or have any questions?

### Follow-up Questions
1. How would the solution change if point C was east of P instead of west?
2. What would the bearing of A and B be if they were southeast of P?
3. How would you calculate the distance between A and B if they were not due north of point C?
4. What trigonometric principles are involved in solving this problem?
5. How would the problem change if P was at a different location, say, due south of C?

### Tip
When dealing with bearings, always convert them to angles measured from the north in the counterclockwise direction to simplify calculations.

Learn how to calculate the distance between points A and B using trigonometry, given their bearings from point P. Understand how to use the tangent function to find horizontal distances. Suitable for Grades 10-12.

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