B is 60km away from A on the bearing of 135:C is 80km away from A in a bearing of 225
1)Find the distance of C from B
2)Bearing of C from B
To find the distance between points B and C, we can use the distance formula. Let's assume that A is the origin (0,0).
The coordinates of point B can be found using the distance and bearing given.
Since B is 60km away from A and on a bearing of 135°, we can find the x and y coordinates of B using trigonometry.
x = 60 * cos(135°)
x = -42.42 km
y = 60 * sin(135°)
y = 42.42 km
So, the coordinates of B are (-42.42 km, 42.42 km).
Similarly, we can find the coordinates of C using the distance and bearing given.
Since C is 80km away from A and on a bearing of 225°, we can find the x and y coordinates of C using trigonometry.
x = 80 * cos(225°)
x = -56.57 km
y = 80 * sin(225°)
y = -56.57 km
So, the coordinates of C are (-56.57 km, -56.57 km).
Using the distance formula, we can find the distance between points B and C:
distance = √[(x2 - x1)^2 + (y2 - y1)^2]
distance = √[(-42.42 - (-56.57))^2 + (42.42 - (-56.57))^2]
distance = √[(-42.42 + 56.57)^2 + (42.42 + 56.57)^2]
distance = √[14.15^2 + 99.99^2]
distance = √[199.94 + 9,997]
distance = √10196.94
distance = 101.98 km
Therefore, the distance between points B and C is approximately 101.98 km.
To find the bearing of C from B, we need to find the angle between the positive x-axis and the line connecting B and C.
We can use trigonometry to find this angle.
Let θ be the angle between the positive x-axis and the line connecting B and C.
tan(θ) = (y2 - y1) / (x2 - x1)
tan(θ) = (-56.57 - 42.42) / (-56.57 - (-42.42))
tan(θ) = -99.99 / -14.15
tan(θ) = 7.07
Taking the arctangent of both sides:
θ = arctan(7.07)
θ ≈ 81.87°
Therefore, the bearing of C from B is approximately 81.87°.
To find the distance of C from B, you can use the formula for finding the distance between two points in a coordinate plane, which is the Pythagorean theorem.
1) Distance of C from B:
Using the Pythagorean theorem, we can find the distance between two points (B and C).
Let's label the coordinates for B as (xB, yB) and the coordinates for C as (xC, yC).
Since B is 60km away from A on a bearing of 135, we can find the coordinates for B by using trigonometry.
Knowing the distance between A and B (60km) and the bearing of 135, we can calculate the change in x and y coordinates as follows:
change in x = distance * cos(bearing) = 60 * cos(135) = -42.43 km (rounded to two decimal places)
change in y = distance * sin(bearing) = 60 * sin(135) = 42.43 km (rounded to two decimal places)
Now, let's assume the coordinates for A are (0, 0).
Therefore, the coordinates for B are (-42.43, 42.43).
Similarly, we can calculate the coordinates for C using the given information that C is 80km away from A in a bearing of 225:
change in x = distance * cos(bearing) = 80 * cos(225) = -56.57 km (rounded to two decimal places)
change in y = distance * sin(bearing) = 80 * sin(225) = -56.57 km (rounded to two decimal places)
The coordinates for C are (-56.57, -56.57).
Now, using the coordinates of B and C, we can find the distance between them:
Distance = √((xC - xB)^2 + (yC - yB)^2) = √((-56.57 - (-42.43))^2 + (-56.57 - 42.43)^2) = √(196.14^2 + (-99.14)^2) = √(38499.6196 + 9829.4596) = √48329.0792 = 219.99 km (rounded to two decimal places)
Therefore, the distance of C from B is approximately 219.99 km.
2) Bearing of C from B:
To find the bearing of C from B, we can use the inverse tangent function to calculate the angle relative to the positive x-axis.
Bearing = atan((yC - yB) / (xC - xB)) = atan((-56.57 - 42.43) / (-56.57 - (-42.43))) = atan(-99.14 / -14.14) = atan(7) = 81.87° (rounded to two decimal places)
Therefore, the bearing of C from B is approximately 81.87°.