Points A and B are on opposite sides of a lake. A point C is 81.3 meters from A. The measure of angle BAC is 78.33°, and the measure of angle ACB is determined to be 34.167°. Find the distance between points A and B (to the nearest meter)

To find the distance between points A and B, we can use the Law of Sines, which is suitable for non-right triangles.

Given:
- \( AC = 81.3 \) meters
- \( \angle BAC = 78.33^\circ \)
- \( \angle ACB = 34.167^\circ \)

First, let's find \( \angle ABC \):
\[ \angle ABC = 180^\circ - \angle BAC - \angle ACB \]
\[ \angle ABC = 180^\circ - 78.33^\circ - 34.167^\circ \]
\[ \angle ABC = 67.503^\circ \]

Now, we can apply the Law of Sines:
\[ \frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(\angle ABC)} \]

Rearranging to solve for \( AB \):
\[ AB = \frac{AC \cdot \sin(\angle ACB)}{\sin(\angle ABC)} \]

Substitute the known values:
\[ AB = \frac{81.3 \cdot \sin(34.167^\circ)}{\sin(67.503^\circ)} \]

Calculate the sines:
\[ \sin(34.167^\circ) \approx 0.5610 \]
\[ \sin(67.503^\circ) \approx 0.9219 \]

Now, compute \( AB \):
\[ AB = \frac{81.3 \cdot 0.5610}{0.9219} \]
\[ AB \approx \frac{45.613}{0.9219} \]
\[ AB \approx 49.47 \]

Rounding to the nearest meter, the distance between points A and B is approximately:
\[ AB \approx 49 \text{ meters} \]

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:

1. How is the Law of Sines derived?
2. What are the conditions for using the Law of Sines?
3. Can the Law of Cosines be used to solve this problem? How?
4. What is the significance of each angle in a triangle?
5. How do you find the area of a triangle using trigonometric functions?
6. What other methods can be used to solve non-right triangles?
7. How can the Law of Sines be applied in real-world scenarios?
8. What are some common mistakes when using the Law of Sines?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when calculating trigonometric functions.

Learn how to calculate the distance between points A and B across a lake using the Law of Sines. Given angles and distances are used to determine the distance AB, rounding to the nearest meter. Suitable for high school students studying trigonometry and triangle properties.

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