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To find the area of triangle \( \Delta EFG \), you can use the **Area Rule**, which states:

\[
\text{Area} = \frac{1}{2} \times EF \times FG \times \sin(\angle ZF)
\]

Here, the given values are:
- \( EF = 1.6 \, \text{km} \)
- \( FG = 2.1 \, \text{km} \)
- \( \angle ZF = 112^\circ \)

### Step-by-Step Calculation:

1. **Convert the angle to radians (optional)**: The angle \( 112^\circ \) is given in degrees, so you can either keep it in degrees or convert it to radians. We will keep it in degrees as most calculators can handle this directly.

2. **Calculate the sine of the angle**: Use the sine function to find \( \sin(112^\circ) \).

3. **Apply the formula**:

\[
\text{Area} = \frac{1}{2} \times 1.6 \times 2.1 \times \sin(112^\circ)
\]

Let's calculate this.The area of triangle \( \Delta EFG \) is approximately \( 1.56 \, \text{km}^2 \).

Do you want more details or have any questions about this calculation?

Here are five related questions to explore further:

1. What happens to the area if the angle \( \angle ZF \) changes?
2. How would you calculate the perimeter of triangle \( \Delta EFG \)?
3. What is the significance of the sine function in this formula?
4. How can you verify the result using a different method, such as the Law of Cosines?
5. How does the area rule compare with Heron's formula for calculating the area of a triangle?

**Tip:** When dealing with angles, ensure your calculator is in the correct mode (degrees or radians) to avoid errors in trigonometric calculations.

Explore how to calculate the area of triangle EFG using the Area Rule. Given side lengths EF = 1.6 km, FG = 2.1 km, and angle ZF = 112 degrees, learn the step-by-step process with trigonometric calculations. Suitable for high school students studying geometry and trigonometry.

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