https://file.aiquickdraw.com/mathbot/file/1729191149490myyudimd.jpg

The image shows a triangle \( ABC \) with the following information:

- \( \angle A = 81^\circ \)
- \( \angle B = 48^\circ \)
- \( BC = 10.8 \) (side opposite \( \angle A \))

The task is to calculate the length of side \( AC \).

### Step 1: Calculate \( \angle C \)
Since the sum of the angles in a triangle is \( 180^\circ \), we can find \( \angle C \) as follows:

\[
\angle C = 180^\circ - 81^\circ - 48^\circ = 51^\circ
\]

### Step 2: Use the Law of Sines
The Law of Sines relates the sides of a triangle to the sines of its angles:

\[
\frac{AC}{\sin B} = \frac{BC}{\sin A}
\]

Substitute the known values:

\[
\frac{AC}{\sin 48^\circ} = \frac{10.8}{\sin 81^\circ}
\]

### Step 3: Solve for \( AC \)
Now, calculate the sine values and solve for \( AC \):

\[
\frac{AC}{0.7431} = \frac{10.8}{0.9877}
\]

\[
\frac{AC}{0.7431} = 10.93
\]

Multiply both sides by 0.7431 to isolate \( AC \):

\[
AC = 10.93 \times 0.7431 = 8.12
\]

### Final Answer:
The length of side \( AC \) is approximately **8.12 units**.

Would you like more details or further clarification? Here are 5 related questions you might find helpful:

1. How do we apply the Law of Cosines in a similar situation?
2. What is the significance of using the Law of Sines for non-right triangles?
3. How can you check if your answer is reasonable when solving triangles?
4. How would this calculation change if the triangle was a right triangle?
5. Can the Law of Sines be applied to obtuse-angled triangles?

**Tip**: Always double-check your sine and angle calculations, especially when using approximations for trigonometric functions!

Calculate the length of side AC in a triangle where angle A = 81°, angle B = 48°, and BC = 10.8.

Learn how to calculate the length of side AC in a triangle with angles A = 81°, B = 48°, and side BC = 10.8 units using the Law of Sines. This problem covers trigonometry concepts and is suitable for high school students.

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