To solve this geometry problem involving triangle PQR:

Part (a) – Finding the length of QS:

Given that QS is perpendicular to PR, we can use trigonometry to find the length of QS. The triangle PQR contains a right triangle, with ∠PQR = 30° and PQ = 20 cm, and SR = 9 cm.

Since QS is perpendicular to PR, we can use trigonometric relationships to find QS. We will use the sine function for this: $\sin(\angle PQR) = \frac{QS}{PQ}$ Substituting the known values: $\sin(30°) = \frac{QS}{20}$ $0.5 = \frac{QS}{20}$ Solving for QS: $QS = 0.5 \times 20 = 10 \, \text{cm}$

Part (b) – Finding the angle x:

We need to find the value of angle ∠SQR = x°. Since the sum of the angles in triangle PQR must equal 180°, we already know: $\angle PQR = 30° \quad \text{and} \quad \angle QRS = 90°$ Therefore, the remaining angle ∠SQR = x° can be calculated as: $x = 180° - \angle PQR - \angle QRS = 180° - 30° - 90° = 60°$

Final Answers:

a) The length of QS is 10 cm.
b) The angle x is 60°.

Let me know if you'd like more details or if you have other questions!

Here are 5 related questions to explore further:

1. How would the solution change if the angle at PQR was different?
2. What if QS was not perpendicular to PR? How would you approach the problem?
3. How can we use the cosine or tangent functions in this type of triangle?
4. What is the significance of using trigonometric identities in solving geometric problems?
5. How does the Pythagorean Theorem relate to this triangle?

Tip: Always check if there are right angles or other known angle properties before applying trigonometric functions!