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!