Find the length of side a. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
aequals
  
enter your response here mi
​(Round to the nearest mile as​ needed.)
B.
There is no solution.

To find the length of side \( a \) using the Law of Sines, we use the following formula:

\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]

Given:
\[ c = 8 \text{ mi} \]
\[ A = 107.73^\circ \]
\[ B = 39.78^\circ \]
\[ C = 32.49^\circ \]

We need to find \( a \). Using the Law of Sines, we can write:

\[
\frac{a}{\sin(A)} = \frac{c}{\sin(C)}
\]

Plugging in the known values:

\[
\frac{a}{\sin(107.73^\circ)} = \frac{8}{\sin(32.49^\circ)}
\]

We need to find the sine values:

\[
\sin(107.73^\circ) \approx 0.9563
\]
\[
\sin(32.49^\circ) \approx 0.5360
\]

Now, plug these into the equation:

\[
\frac{a}{0.9563} = \frac{8}{0.5360}
\]

Solving for \( a \):

\[
a = \frac{8 \cdot 0.9563}{0.5360} \approx \frac{7.6504}{0.5360} \approx 14.28 \text{ mi}
\]

Rounding to the nearest mile:

\[
a \approx 14 \text{ mi}
\]

Therefore, the length of side \( a \) is approximately:
A. \( a = 14 \) mi

Would you like any more details or have any questions about the process?

Here are 5 questions you might consider asking:
1. How do you find the length of side \( b \) in this triangle?
2. Can you explain how to solve a triangle using the Law of Cosines?
3. What are the steps for solving a triangle given all three sides?
4. How can you determine if a triangle is valid based on its angles and sides?
5. What are the common mistakes to avoid when using the Law of Sines?

**Tip:** When solving trigonometric problems, make sure your calculator is set to the correct mode (degrees or radians) based on the given angles.

Learn how to find the length of side a in a triangle using the Law of Sines with detailed steps and calculations. Given angles and side lengths are used to determine the unknown side length. Suitable for high school students studying trigonometry.

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