city a is 300km due east of city b.city c is 200km on a bearing of 123 degrees from city b.how far is it from c to a

use the law of cosines.

AC^2 = 300^2 + 200^2 - 2*300*200 cos 33°
AC = 171

I need the diagram of it please

I need a diagram of the bearing

A city A is 300km due east of city B. City C is 200km on a bearing of 120 degrees from city B. so how far is city C from city A?

To find the distance from city C to city A, we can use the concept of vector addition. Let's break down the problem into steps:

Step 1: Determine the displacement from city B to city C.
- City C is 200 km away from city B on a bearing of 123 degrees.
- The displacement can be represented as a vector with magnitude (200 km) and direction (123 degrees).

Step 2: Determine the displacement from city B to city A.
- City A is 300 km due east of city B.
- Since it is due east, the displacement is in the east direction only.
- The displacement can be represented as a vector with magnitude (300 km) and direction (0 degrees).

Step 3: Add the two displacements together to get the total displacement from city C to city A.
- Vector addition involves adding the magnitudes and summing up the directions.
- To add the vectors, we can break them down into their horizontal (x-component) and vertical (y-component) parts.
- The x-component represents the east/west direction, and the y-component represents the north/south direction.

Let's calculate the x and y components for each vector:

For the displacement from B to C:
- x-component: 200 km * cos(123 degrees)
- y-component: 200 km * sin(123 degrees)

For the displacement from B to A:
- x-component: 300 km * cos(0 degrees)
- y-component: 300 km * sin(0 degrees)

Step 4: Add the x-components and y-components separately to get the total displacement from city C to city A.
- The total displacement (dx, dy) = (x-component from B to C + x-component from B to A, y-component from B to C + y-component from B to A).

Step 5: Calculate the magnitude of the total displacement.
- The magnitude of the total displacement can be found using the Pythagorean theorem:
- Magnitude = sqrt(dx^2 + dy^2)

Now, let's calculate the distance from city C to city A:

Step 1: Calculate the x-component of the displacement from B to C:
x-component = 200 km * cos(123 degrees)
x-component = -126.94 km (rounded to two decimal places)

Step 2: Calculate the y-component of the displacement from B to C:
y-component = 200 km * sin(123 degrees)
y-component = 112.59 km (rounded to two decimal places)

Step 3: Calculate the x-component of the displacement from B to A:
x-component = 300 km * cos(0 degrees)
x-component = 300 km

Step 4: Calculate the y-component of the displacement from B to A:
y-component = 300 km * sin(0 degrees)
y-component = 0 km

Step 5: Calculate the total displacement from city C to city A:
dx = -126.94 km + 300 km = 173.06 km (rounded to two decimal places)
dy = 112.59 km + 0 km = 112.59 km (rounded to two decimal places)

Step 6: Calculate the magnitude of the total displacement:
magnitude = sqrt(dx^2 + dy^2)
magnitude = sqrt((173.06 km)^2 + (112.59 km)^2)
magnitude ≈ 206.08 km (rounded to two decimal places)

Therefore, the distance from city C to city A is approximately 206.08 km.