B is 60km from A on the bearing of 135 degree ,C is 80km away from A in a bearing of 225 degree find the (I)Distance of C from B
(II)Bearing of C from B
I) To find the distance of C from B, we can use the Law of Cosines.
Let's denote the distance from C to B as x.
Using the Law of Cosines:
x² = 60² + 80² - 2(60)(80)cos(135°)
Simplifying:
x² = 3600 + 6400 - 9600cos(135°)
x² = 10000 - 9600(-0.7071)
x² = 10000 + 6787.584
x² ≈ 16787.584
x ≈ √16787.584
x ≈ 129.51 km
Therefore, the distance of C from B is approximately 129.51 km.
II) To find the bearing of C from B, we can use the formula:
Bearing of C from B = Bearing of C from A - Bearing of B from A
The bearing of C from A is 225° and the bearing of B from A is 135°.
Bearing of C from B = 225° - 135°
Bearing of C from B = 90°
Therefore, the bearing of C from B is 90°.
To find the distance of C from B, we can make use of the Law of Cosines.
First, let's visualize the situation:
A
/\
/ \
B ---- C
Given:
- Distance between points A and B: AB = 60 km
- Bearing from A to B: 135°
- Distance between points A and C: AC = 80 km
- Bearing from A to C: 225°
To find the distance of C from B (CB), we need to make use of the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the side opposite the angle C.
In this case:
- AB = a
- AC = b
- CB = c
- angle C = angle BAC ('angle from B to A to C')
(I) Distance of C from B:
Using the Law of Cosines, we have:
CB^2 = AB^2 + AC^2 - 2 * AB * AC * cos(angle BAC)
Substituting the given values:
CB^2 = 60^2 + 80^2 - 2 * 60 * 80 * cos(angle BAC)
Now, let's find angle BAC:
angle BAC = 360° - (135° + 225°)
angle BAC = 360° - 360°
angle BAC = 0°
Substituting angle BAC = 0° into the equation:
CB^2 = 60^2 + 80^2 - 2 * 60 * 80 * cos(0°)
CB^2 = 3600 + 6400 - 2 * 4800 * 1
CB^2 = 3600 + 6400 - 9600
CB^2 = 4000
Taking the square root of both sides:
CB = √4000
CB = 63.24 km (approx.)
Therefore, the distance of C from B is approximately 63.24 km.
(II) To find the bearing of C from B, we subtract the bearing of A from B (135°) from the bearing of A from C (225°):
Bearing of C from B = Bearing of A from C - Bearing of A from B
Bearing of C from B = 225° - 135°
Bearing of C from B = 90°
Therefore, the bearing of C from B is 90°.