Establish a coordinate system.

To solve this problem, we will use trigonometry to determine the distances and bearings, which is the analytical equivalent of a precise scale drawing.

Step 1: Establish a coordinate system.
Let Abeokuta (Abc) be at the origin $(0,0)$. We define North as the positive y-axis and East as the positive x-axis.

Step 2: Determine the coordinates of Ikeja (IK).
Ikeja is 60 km due South of Abeokuta.
Therefore, the coordinates of Ikeja are $(0, -60)$.

Step 3: Calculate the displacement of Ibadan (Ib) from Ikeja.
Ibadan is 110 km from Ikeja on a bearing of $036^\circ$. A bearing is measured clockwise from the North line.
The eastward displacement (x-component) from Ikeja is given by $110 \sin(36^\circ)$.
The northward displacement (y-component) from Ikeja is given by $110 \cos(36^\circ)$.

$$\text{Eastward displacement} = 110 \sin(36^\circ) \approx 110 \times 0.5878 \approx 64.658 \text{ km}$$ $$\text{Northward displacement} = 110 \cos(36^\circ) \approx 110 \times 0.8090 \approx 88.990 \text{ km}$$

Step 4: Solve part b) - How far north and how far east Ibadan is from Ikeja.
Based on the calculations in Step 3:

• Ibadan is approximately 89.0 km North of Ikeja.
• Ibadan is approximately 64.7 km East of Ikeja.

Step 5: Determine the absolute coordinates of Ibadan (Ib) relative to Abeokuta.
To find the coordinates of Ibadan relative to Abeokuta, we add the displacement from Ikeja to Ikeja's coordinates:

$$x_{\text{Ibadan}} = x_{\text{Ikeja}} + \text{Eastward displacement} = 0 + 64.658 = 64.658 \text{ km}$$ $$y_{\text{Ibadan}} = y_{\text{Ikeja}} + \text{Northward displacement} = -60 + 88.990 = 28.990 \text{ km}$$

So, Ibadan's coordinates relative to Abeokuta are approximately $(64.658, 28.990)$.

Step 6: Solve part a) - The bearing and distance of Ibadan from Abeokuta.
The distance of Ibadan from Abeokuta is the hypotenuse of the right-angled triangle formed by its coordinates:

$$\text{Distance} = \sqrt{x_{\text{Ibadan}}^2 + y_{\text{Ibadan}}^2}$$ $$\text{Distance} = \sqrt{(64.658)^2 + (28.990)^2}$$ $$$$