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To find the value of \( x \) in this triangle, we can use the *Cosine Rule*, as we have two sides and the included angle between them (triangle with angles 65°, 33°, and 37°).

### Step-by-Step Solution

#### Step 1: Verify Triangle Structure
We have a triangle with:
- Side lengths of 8.1 cm and 4.5 cm.
- Angles of 33°, 65°, and 37°.

Given that this is a non-right triangle, we’ll apply the Cosine Rule to find side \( x \).

#### Step 2: Cosine Rule
The Cosine Rule states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where:
- \( c \) is the side opposite the angle \( C \),
- \( a \) and \( b \) are the other two sides, and
- \( C \) is the angle between sides \( a \) and \( b \).

In our case:
- \( a = 8.1 \) cm,
- \( b = 4.5 \) cm,
- \( C = 65^\circ \),
- \( x \) is the side opposite the 65° angle.

#### Step 3: Substitute Values
Using the formula:
\[
x^2 = 8.1^2 + 4.5^2 - 2 \cdot 8.1 \cdot 4.5 \cdot \cos(65^\circ)
\]

Calculating each term:

1. \( 8.1^2 = 65.61 \)
2. \( 4.5^2 = 20.25 \)
3. \( 2 \cdot 8.1 \cdot 4.5 = 72.9 \)
4. \(\cos(65^\circ) \approx 0.4226\)

So,
\[
x^2 = 65.61 + 20.25 - 72.9 \cdot 0.4226
\]

Calculating further:
\[
x^2 = 65.61 + 20.25 - 30.8
\]
\[
x^2 = 55.06
\]
\[
x = \sqrt{55.06} \approx 7.42 \, \text{cm}
\]

### Answer
The value of \( x \) is approximately \( 7.42 \) cm, correct to two decimal places.

Would you like a further breakdown or have any questions?

Here are some related questions:
1. How would we approach this problem if one of the sides was unknown instead of the angle?
2. How does the Cosine Rule differ from the Sine Rule, and when is each used?
3. What steps are needed to verify the accuracy of angle measurements in such problems?
4. What if the angle given was obtuse instead of acute; would it change our approach?
5. How does knowing two sides and an opposite angle help in solving non-right triangles?

**Tip:** Always verify the triangle’s type (acute, obtuse, or right) to choose the correct trigonometric rule effectively.

Find the value of x in the triangle with sides 8.1 cm, 4.5 cm, and angle 65° between them.

Learn how to find the unknown side length of a triangle using the Cosine Rule. Given two sides (8.1 cm, 4.5 cm) and an included angle (65°), this example walks through the process to determine side x. Suitable for high school students studying trigonometry.

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