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This diagram appears to be a triangle with angles and distances labeled. The given information includes:

- Angle \( \angle TQP = 30^\circ \)
- Angle \( \angle PTO = 18^\circ \)
- Distance \( PQ = 30 \text{ meters} \)
- Height \( h \) from point \( T \) perpendicular to line \( PQ \), with \( O \) being the point where the height intersects \( PQ \).

### To solve for the height \( h \):

We can use trigonometric relationships to find \( h \). Here's how we proceed:

1. Since \( \angle TQP = 30^\circ \), and \( O \) is the point where the perpendicular from \( T \) meets \( PQ \), we can apply trigonometric functions in triangle \( TQO \).

   In the right triangle \( TQO \), we know that:
   \[
   \sin(30^\circ) = \frac{h}{TQ}
   \]

2. First, we need to find the length of \( TQ \). To do this, we can apply the Law of Sines in \( \triangle PQT \) using the angle \( \angle PTO = 18^\circ \), the given side \( PQ = 30 \text{ meters} \), and other known angles.

I will now calculate step-by-step and find \( h \).The length of side \( TQ \) is approximately \( 18.54 \, \text{m} \), and the height \( h \) from point \( T \) to line \( PQ \) is approximately \( 9.27 \, \text{m} \).

Would you like further details on the steps or explanations? Here are five related questions you might find helpful:

1. How would the height \( h \) change if the angle \( \angle TQP \) were larger?
2. What would be the effect on the triangle if the length of \( PQ \) were increased?
3. How can the Law of Cosines be used to verify these results?
4. What if the angle \( \angle PTO \) was different — how would it affect the solution?
5. Could we solve this problem using coordinate geometry or vectors?

**Tip:** When solving triangles, always check if using the Law of Sines or Cosines is more appropriate based on the known angles and sides.

Solve the height of the triangle based on the given angles and distances.

Learn how to solve for the height of a triangle using trigonometric ratios and the Law of Sines. This problem involves angles of 30° and 18°, and a base length of 30 meters, providing a step-by-step solution to find the height.

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