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
Let A be (0,0).
B is then (0,10)
C is 10 km at 060 from B.
Bearings are measured from north in a clockwise direction, which means that C is 10 km at 30° north of east.
So C is at
(0+10cos(30), 10+10sin(30))
=(10+(√3)/2 , 10+10/2)
=(10+(√3)/2 , 15)
To find the bearing and distance of Point C from Point A, we need to use the information given and apply some basic trigonometry.
Given:
- Point B is 10 km north of Point A.
- Point C is 10 km from Point B on a bearing of 060 degrees from B.
First, let's determine the coordinates of Point B and Point C.
Let's assume Point A is at coordinates (0, 0).
Since Point B is 10 km north of Point A, the coordinates of Point B would be (0, 10).
To find the coordinates of Point C, we need to use trigonometry. We know that Point C is 10 km from B on a bearing of 060 degrees from B.
The bearing of 060 degrees means that the angle between the line BC and the positive x-axis is 60 degrees.
Using trigonometric ratios, we can find the change in x and change in y between B and C.
Change in x = distance * cos(theta)
Change in y = distance * sin(theta)
In this case, distance = 10 km and theta = 60 degrees.
Change in x = 10 km * cos(60) = 10 km * 0.5 = 5 km
Change in y = 10 km * sin(60) = 10 km * (√3/2) ≈ 8.66 km
Now, we can calculate the coordinates of Point C.
Starting from B (0, 10), we move 5 km to the right (positive x-axis) and 8.66 km upwards (positive y-axis).
Coordinates of Point C = (5 km, 10 km + 8.66 km) = (5, 18.66)
Now, we can find the distance between A and C using the Pythagorean theorem.
Distance AC = √((x2 - x1)^2 + (y2 - y1)^2)
Distance AC = √((5 - 0)^2 + (18.66 - 0)^2)
Distance AC = √(25 + 347.9556)
Distance AC = √372.9556
Distance AC ≈ 19.32 km
To find the bearing of C from A, we need to find the angle between the positive x-axis and the line AC.
Bearing of C from A = arctan(change in y / change in x)
Bearing of C from A = arctan(8.66 km / 5 km)
Bearing of C from A ≈ 59.04 degrees
Therefore, the bearing of C from A is approximately 59.04 degrees, and the distance between A and C is approximately 19.32 km.
To find the bearing and distance of point C from point A, we can use vector addition and trigonometry.
1. Determine the position of point B relative to point A:
Point B is 10 km north of Point A. Since north is directly up on a map, we can say that the displacement from A to B is 10 km in the y-direction.
2. Determine the position of point C relative to point B:
Point C is 10 km from B on a bearing of 060. A bearing of 060 means that the angle between the direction from B to C and the positive x-axis is 60 degrees. Using trigonometry, we can decompose this displacement into its x and y components.
The x-component is given by 10 km * cos(60°) = 5 km.
The y-component is given by 10 km * sin(60°) = 8.66 km.
3. Calculate the position of point C:
To find the position of point C relative to point A, we need to add the position of B relative to A with the position of C relative to B.
The x-position of C relative to A is the sum of the x-positions:
x_C = x_B + x_CB = 0 + 5 km = 5 km.
The y-position of C relative to A is the sum of the y-positions:
y_C = y_B + y_CB = 10 km + 8.66 km = 18.66 km.
4. Calculate the distance between points A and C:
The distance between two points can be found using the Pythagorean theorem. In this case, the distances in the x and y directions form a right triangle.
Distance_AC = sqrt((x_C)^2 + (y_C)^2)
= sqrt((5 km)^2 + (18.66 km)^2)
= sqrt(25 km^2 + 348.16 km^2)
= sqrt(373.16 km^2)
≈ 19.32 km.
5. Calculate the bearing of C from A:
The bearing between two points can be found using trigonometry. In this case, we can find the angle between the direction from A to C and the positive x-axis.
Bearing_AC = atan(y_C / x_C)
= atan(18.66 km / 5 km)
≈ 75.96°
However, since point C is positioned to the right of point A, we need to adjust the bearing by adding 180 degrees.
Bearing_AC = 75.96° + 180°
≈ 255.96°.
Therefore, the bearing of point C from point A is approximately 255.96 degrees and the distance is approximately 19.32 km.