A plane leaves its airport on the east coast of country A and it flies in a direction of 83degrees. Because of bad weather, it returns to another airport 210km due north of its home base. To get to the new airport, it flies in a direction of 280degrees. What is the total distance the airplane flew?

To determine the total distance the airplane flew, let’s break down the problem step by step:

1. **Initial Leg of the Flight (Direction 83°)**:
   - The plane leaves its home airport and flies a certain distance in the direction of \(83^\circ\) (nearly eastward but slightly north).

2. **Second Leg of the Flight (Return Direction 280°)**:
   - Due to weather, the plane has to turn back and flies in the direction of \(280^\circ\) to reach a new airport that is \(210 \, \text{km}\) north of its starting point.

To solve for the total distance, we need to find:
1. The distance flown in the first leg (denote it as \(d\)),
2. The distance flown in the second leg (denote it as \(D\)).

### Step 1: Set Up Coordinates
Let the starting airport be the origin point, \((0,0)\).

1. Since the new airport is located \(210 \, \text{km}\) north of the starting point, its coordinates are \((0, 210)\).

### Step 2: Resolve First Leg into Coordinates
The plane's initial direction is \(83^\circ\). For this leg, the coordinates of the endpoint after flying distance \(d\) can be calculated by:
   \[
   x_1 = d \cdot \cos(83^\circ)
   \]
   \[
   y_1 = d \cdot \sin(83^\circ)
   \]

### Step 3: Resolve Second Leg into Coordinates
In the second leg, the plane flies in the direction of \(280^\circ\) to reach the point \((0, 210)\). The angle \(280^\circ\) is equivalent to \(-80^\circ\) (or \(360^\circ - 280^\circ\)), so:
   \[
   x_2 = x_1 + D \cdot \cos(280^\circ)
   \]
   \[
   y_2 = y_1 + D \cdot \sin(280^\circ)
   \]


A plane leaves its airport on the east coast of country A and it flies in a direction of 83 degrees. Because of bad weather, it returns to another airport 210 km due north of its home base. To get to the new airport, it flies in a direction of 280 degrees. What is the total distance the airplane flew?

Determine the total distance a plane flew after taking off at 83 degrees and returning at 280 degrees, using trigonometric and coordinate geometry techniques. The plane returns 210 km north of its origin. This solution includes calculations of trigonometric components and distances, suitable for Grades 10-12.

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