City y is 200km and in the direction of 048 from city x. City x is 80km and in the direction of 108 from city y. Find the distance of city z from city x and the bearing of city x from z

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To find the distance of city Z from city X, we can use the Law of Cosines in the triangle formed by cities X, Y, and Z.

Let's denote the distance from city X to city Z as d and the angle between the line connecting cities X and Y and the line connecting cities X and Z as θ.

Using the Law of Cosines:
d^2 = (200 km)^2 + (80 km)^2 - 2 * 200 km * 80 km * cos(θ)

Now, let's find the angle θ:
θ = 360° - 048° - 108° = 204°

Substituting the values into the equation:
d^2 = (200 km)^2 + (80 km)^2 - 2 * 200 km * 80 km * cos(204°)

Simplifying this equation will give us the distance of city Z from city X.

Now, let's find the bearing of city X from city Z.

First, we need to find the bearing from city Y to city X, which is given as 108°.

The bearing from city X to city Z will be the opposite direction of the bearing from city Y to city X. Therefore, the bearing from city X to city Z will be 288° (108° + 180°).

Now, we have the distance of city Z from city X and the bearing of city X from city Z.

I assume you meant

City z is 80km and in the direction of 108 from city y

So, with x at (0,0)
y is at (133.8,148.6)
z is at y + (76.1,-24.7) = (209.9,123.9)

Now just apply the usual distance formula and conversion to polar form.