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To find the length of side \( AC \) in triangle \( ABC \), we can use the **Cosine Rule** because we are given two sides and the included angle.

The **Cosine Rule** is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Where:
- \( c \) is the side opposite the angle \( C \) (in this case, side \( AC \)),
- \( a \) and \( b \) are the other two sides (here \( a = 8.2 \) cm, \( b = 13.5 \) cm),
- \( C \) is the included angle (here \( C = 81^\circ \)).

Let's substitute these values into the formula and calculate the length of side \( AC \).

\[
AC^2 = 8.2^2 + 13.5^2 - 2 \times 8.2 \times 13.5 \times \cos(81^\circ)
\]

I'll calculate this now.The length of side \( AC \) is approximately \( 14.66 \) cm.

Would you like further details or explanations on this? 

Here are 5 related questions to explore:

1. How would the answer change if the angle was smaller, say 60°?
2. What other methods could be used if all angles were known instead of sides?
3. Can the Cosine Rule be used for non-obtuse triangles as well?
4. What is the difference between the Sine Rule and Cosine Rule?
5. How do we know when to apply the Cosine Rule versus the Pythagorean theorem?

**Tip:** When using trigonometric formulas like the Cosine Rule, always ensure that angles are in the correct unit (degrees or radians) for calculations.

The diagram shows triangle ABC. Given AB = 13.5 cm, AC = ?, BC = 8.2 cm, and angle BAC = 81°. Calculate the length of side AC.

This example explains how to calculate the length of side AC in triangle ABC, using the Cosine Rule. Given AB = 13.5 cm, BC = 8.2 cm, and angle BAC = 81°, the length of AC is approximately 14.66 cm. Suitable for students in Grades 9-12.

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