https://r2.erweima.ai/mathbot/image/17261866423104oxudi8a.jpg

In the given image, a triangle is formed with the following elements:

- \( AB \), the length between points \( A \) and \( B \), is 23.7 m.
- \( AC \), the length between points \( A \) and \( C \), is 37.8 m.
- \( \delta = 54^\circ \), the angle at point \( C \).

To solve or find further properties about this triangle, we can use trigonometric rules, particularly the Law of Cosines or the Law of Sines, depending on what you are looking to find.

### To solve for side \( BC \):
We can use the **Law of Cosines**:

\[
BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\delta)
\]

Substituting the known values:

\[
BC^2 = (23.7)^2 + (37.8)^2 - 2 \cdot (23.7) \cdot (37.8) \cdot \cos(54^\circ)
\]

Let me calculate this for you.The length of side \( BC \) is approximately 30.62 meters.

Would you like to explore further details about this triangle or any other aspects? Here are some follow-up questions that can expand on this topic:

1. Would you like to calculate the area of the triangle \( ABC \)?
2. Do you need the remaining angles in the triangle?
3. Should we calculate the perimeter of the triangle?
4. Do you want to verify the result using the Law of Sines?
5. Would you like to explore how the Law of Cosines works for different triangle configurations?

**Tip:** The Law of Cosines is particularly useful when you know two sides and the included angle of a triangle, as seen in this problem.

Given a triangle with side AB = 23.7 m, side AC = 37.8 m, and the angle at C, δ = 54°, find side BC using the Law of Cosines.

In this example, we calculate the side BC of a triangle where AB = 23.7 m, AC = 37.8 m, and the angle δ = 54° using the Law of Cosines. This method is perfect for high school students studying trigonometry.

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