Question

Villages A,B,C,D are such that B is 4km due east of A. C is 3km due south of B and D is 4km S50°W from C. Calculate the distance and bearing of A from D without using coordinate geometry

Answer 1

To find the distance and bearing of village A from village D, we can break down the problem step by step, using trigonometry and the properties of triangles.

1. **Positioning the Villages:**
- Start by placing village A at point \( A \).
- Village B is 4 km due east of A. Therefore, position B at \( B(4, 0) \) in a hypothetical coordinate system (to aid visualization).
- Village C is 3 km due south of B. Thus, the position of C will be \( C(4, -3) \) as you move south (downward).

2. **Positioning Village D:**
- D is located 4 km S50°W from C.
- To analyze this direction, S50°W means you move 50° towards the west (from south).
- The angle measured clockwise from the north is 180° + 50° = 230°.

3. **Finding the components for D:**
- To find D using the angle, we calculate the directional components:
- The angle S50°W gives us:
\[
\text{For D's movement:}
\]
\[
\text{East-West (x-axis): } -4 \cos(50° \text{ in the negative direction east}) = -4 \times \cos(50°)
\]
\[
\text{North-South (y-axis): } -4 \sin(50°) \text{ (since it moves south)}
\]

- Now plugging in the 4 km distance:
\[
\text{For the east-west (x-direction): } -4 \cos(50°) = -4 \times 0.643 = -2.572 \text{ km}
\]
\[
\text{For the north-south (y-direction): } -4 \sin(50°) = -4 \times 0.766 = -3.064 \text{ km}
\]

4. **Calculating the exact position of D relative to C:**
- The coordinates of D relative to C:
\[
D_x = 4 - 2.572 = 1.428 \text{ km}
\]
\[
D_y = -3 - 3.064 = -6.064 \text{ km}
\]
- Thus, the coordinates of D are approximately \( (1.428, -6.064) \) relative to the origin A.

5. **Calculating the position of D relative to A:**
- With A at (0, 0), relative position of D is \( D(1.428, -6.064) \).
- To find the distance \( AD \):
\[
AD = \sqrt{ (1.428 - 0)^2 + (-6.064 - 0)^2}
\]
\[
= \sqrt{ 1.428^2 + 6.064^2 }
\]
\[
= \sqrt{ 2.040384 + 36.797696 } = \sqrt{38.83808} \approx 6.22 \text{ km}
\]

6. **Finding the bearing of A from D:**
- The angle \( \theta \) from the east to AD can be found using:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-6.064}{1.428}
\]
- \( \theta = \tan^{-1}(-4.25) \). This gives an angle of approximately \( \tan^{-1}(4.25) \approx -77.78° \) from the east.
- Adjusting for bearing (from North through east):
\[
\text{Bearing} = 180° + \theta \approx 180° - 77.78° \text{ (since we measure clockwise from north)}
\]
\[
\approx 102.22° \text{ S78.22°E } \text{(since we're below the x-axis)}
\]

In conclusion, the distance from A to D is approximately:

**Distance \( AD \approx 6.22 \text{ km} \)**

**Bearing of A from D \( \approx 102.22° \) (or S78.22°E)**.