he bearings of the lines OA, OB, OC, OD are 30°30 ′, 140°15 ′, 220°45 ′ and 310°30 ′, respectively. Find the angles ∠AOB, ∠BOC and ∠COD.
The fore bearings of the lines AB, BC, CD and DE, are 45°30 ′ 120°15 ′, 200°30 ′ and 280°45 ′, respectively. Find angles ∠B, ∠C and ∠D.
The Correct Answer and Explanation is :
To solve the given problems, we need to calculate the angles between the lines from the bearings provided, first for the angles ∠AOB, ∠BOC, ∠COD, and then for the angles ∠B, ∠C, and ∠D.
1. Angles ∠AOB, ∠BOC, ∠COD (from the bearings of OA, OB, OC, and OD):
We are given the following bearings:
- OA: 30°30′
- OB: 140°15′
- OC: 220°45′
- OD: 310°30′
To find the angles between the lines, we calculate the difference between the bearings of the respective lines:
∠AOB:
The bearing of line OA is 30°30′, and the bearing of line OB is 140°15′. The angle ∠AOB is the difference between these two bearings:
[
\text{∠AOB} = 140°15′ – 30°30′ = 109°45′
]
∠BOC:
The bearing of line OB is 140°15′, and the bearing of line OC is 220°45′. The angle ∠BOC is:
[
\text{∠BOC} = 220°45′ – 140°15′ = 80°30′
]
∠COD:
The bearing of line OC is 220°45′, and the bearing of line OD is 310°30′. The angle ∠COD is:
[
\text{∠COD} = 310°30′ – 220°45′ = 89°45′
]
2. Angles ∠B, ∠C, and ∠D (from the fore bearings of lines AB, BC, CD, and DE):
We are given the following fore bearings:
- AB: 45°30′
- BC: 120°15′
- CD: 200°30′
- DE: 280°45′
To find the angles at points B, C, and D, we calculate the difference between the respective fore bearings:
∠B:
The fore bearing of line AB is 45°30′, and the fore bearing of line BC is 120°15′. The angle ∠B is:
[
\text{∠B} = 120°15′ – 45°30′ = 74°45′
]
∠C:
The fore bearing of line BC is 120°15′, and the fore bearing of line CD is 200°30′. The angle ∠C is:
[
\text{∠C} = 200°30′ – 120°15′ = 80°15′
]
∠D:
The fore bearing of line CD is 200°30′, and the fore bearing of line DE is 280°45′. The angle ∠D is:
[
\text{∠D} = 280°45′ – 200°30′ = 80°15′
]
Summary of Results:
- ∠AOB = 109°45′
- ∠BOC = 80°30′
- ∠COD = 89°45′
- ∠B = 74°45′
- ∠C = 80°15′
- ∠D = 80°15′
Explanation:
- Bearings are the angles measured clockwise from the north direction.
- To find the angle between two lines, we subtract the smaller bearing from the larger bearing.
- If the result exceeds 180°, subtract it from 360° to get the smaller angle.
- These calculations are based on the assumption that we are using the correct direction for each bearing and ensuring that the angle between the lines is between 0° and 180°.
This method ensures that we account for the relative positions of the lines accurately.