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To solve for \( p \), we can use the law of sines since we have a non-right-angled triangle and know two angles (67° and 38°) and one side (16.5 mm). Here's how:

1. **Sum of angles in a triangle**: The third angle in the triangle can be calculated as:
   \[
   180^\circ - 67^\circ - 38^\circ = 75^\circ
   \]
   So, the third angle is 75°.

2. **Using the Law of Sines**: The law of sines states:
   \[
   \frac{p}{\sin(38^\circ)} = \frac{16.5 \text{ mm}}{\sin(75^\circ)}
   \]

3. **Solving for \( p \)**:
   \[
   p = \frac{16.5 \times \sin(38^\circ)}{\sin(75^\circ)}
   \]

Let's calculate this now.The length of \( p \) is approximately **10.5 mm**, rounded to one decimal place.

Would you like further details on the solution or any clarifications?

Here are some related questions you might want to explore:
1. How can you use the Law of Cosines in solving triangle problems?
2. What is the significance of the Law of Sines in non-right-angled triangles?
3. How do we calculate angles in a triangle given two sides and an angle?
4. Can we solve similar problems using trigonometric identities?
5. How can you determine whether a triangle is obtuse, acute, or right-angled based on its angles?

**Tip**: Always ensure that your calculator is in the correct mode (degrees or radians) when working with trigonometric functions.

Calculate the length of p. Give your answer in millimetres to 1 d.p.

Learn how to calculate the length of side p in a triangle using the Law of Sines. This problem involves trigonometric concepts and is suitable for students in Grades 10-12. The given angles are 67° and 38°, with a known side of 16.5 mm.

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