Math Problem Statement
Solution
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: Substituting the known values: Solving for QS:
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: Therefore, the remaining angle ∠SQR = x° can be calculated as:
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:
- How would the solution change if the angle at PQR was different?
- What if QS was not perpendicular to PR? How would you approach the problem?
- How can we use the cosine or tangent functions in this type of triangle?
- What is the significance of using trigonometric identities in solving geometric problems?
- 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!
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Geometry
Right-angle triangles
Formulas
sin(θ) = opposite / hypotenuse
Sum of angles in a triangle = 180°
Theorems
Trigonometric ratios
Angle sum property of triangles
Suitable Grade Level
Grades 9-11