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The question asks you to use the Cosine Rule to calculate the unknown side length \( x \). The triangle has sides 22 and 28, and an included angle of 97°.

The **Cosine Rule** is given by:
\[
x^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where:
- \( x \) is the unknown side,
- \( a = 22 \), \( b = 28 \) are the known sides,
- \( C = 97^\circ \) is the angle between these sides.

### Step-by-step solution:

1. Apply the values into the formula:
\[
x^2 = 22^2 + 28^2 - 2 \times 22 \times 28 \times \cos(97^\circ)
\]

2. Calculate each part:
   - \( 22^2 = 484 \)
   - \( 28^2 = 784 \)
   - \( 2 \times 22 \times 28 = 1232 \)
   - \( \cos(97^\circ) \approx -0.1219 \) (since cosine of an obtuse angle is negative)

3. Substitute these into the equation:
\[
x^2 = 484 + 784 - 1232 \times (-0.1219)
\]
\[
x^2 = 1268 + 150.16
\]
\[
x^2 = 1418.16
\]

4. Take the square root:
\[
x = \sqrt{1418.16} \approx 37.66
\]

So, the unknown side length \( x \approx 37.66 \) units.

Would you like further details on any part? Let me know if you have any questions!

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Here are 5 follow-up questions based on this problem:
1. How would the solution change if the angle was acute, say 70°?
2. What is the geometric significance of the negative cosine value for obtuse angles?
3. Can you derive the Cosine Rule from basic trigonometric principles?
4. How does the Cosine Rule differ from the Sine Rule?
5. What happens if all angles in the triangle are known but none of the sides?

**Tip**: When working with angles in trigonometric equations, always ensure that your calculator is in the correct mode (degrees or radians) based on the angle measurement given.

Use the Cosine rule to calculate the unknown side length. Given: side lengths 22, 28, and included angle 97°.

Learn how to apply the Cosine Rule to solve for an unknown side of a triangle with given sides of 22 and 28, and an included angle of 97°. This example covers key trigonometric concepts suitable for Grades 10-12.

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