Point B is 10 KM north of Point A and Point C is 10 Km from B on a bearing of 060 from b. Find the bearing and distance of C from A
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To find the bearing and distance of C from A, we can break down the problem into steps:
Step 1: Determine the coordinates of points A, B, and C.
Since point B is 10 KM north of point A, we can assume the coordinates of A are (0, 0) and the coordinates of B are (0, 10).
Now, to find the coordinates of C, we need to determine its position relative to B. Given that point C is 10 KM from B on a bearing of 060 from B, we need to find the displacement of C from B in the eastward (x) and northward (y) direction.
Step 2: Calculate the displacement of point C from B.
To find the displacement of C from B, we will use trigonometry.
Given the distance of 10 KM and the bearing of 060 degrees, we can split this into horizontal (x) and vertical (y) components.
x = distance * cos(bearing)
x = 10 * cos(60°) = 10 * 0.5 = 5 KM
y = distance * sin(bearing)
y = 10 * sin(60°) = 10 * √3/2 = 10 * 0.866 = 8.66 KM
Therefore, the coordinates of C relative to B are (5, 8.66).
Step 3: Add the coordinate values of C relative to B to the coordinates of B to find the coordinates of C relative to A.
Adding the coordinate values of C relative to B (5, 8.66) to the coordinates of B (0, 10), we get:
C = (0 + 5, 10 + 8.66) = (5, 18.66)
Now, we have the coordinates of C relative to A, which are (5, 18.66).
Step 4: Calculate the distance of C from A.
To calculate the distance of C from A, we use the distance formula, which is the distance between two points in a coordinate plane.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((5 - 0)^2 + (18.66 - 10)^2)
Distance = sqrt(25 + 87.56)
Distance = sqrt(112.56)
Distance ≈ 10.62 KM
Therefore, the distance of C from A is approximately 10.62 KM.
Step 5: Calculate the bearing of C from A.
To calculate the bearing of C from A, we will use trigonometry.
First, we find the angle between the line joining A and C with the north direction.
θ = arctan((y2 - y1)/(x2 - x1))
θ = arctan((18.66 - 0)/(5 - 0))
θ = arctan(18.66/5)
θ ≈ arctan(3.732)
Next, we convert the angle to the appropriate bearing notation, which is a bearing measured clockwise from the north direction.
Bearing = 90° - θ
Bearing = 90° - arctan(3.732)
Bearing ≈ 90° - 75.96°
Bearing ≈ 14.04°
Therefore, the bearing of C from A is approximately 14.04°.