Question

In the diagram above, triangle PQR has a right angle at Q. What is the ratio of the area of triangle PQS to the area of triangle QRS? 
(1) Line segment \( QS \) is perpendicular to \( PR \) and has a length of 12. 
(2) PQR has a perimeter of 60.

• A. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
• B. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
• C. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In geometry problems, combining multiple pieces of information can often help in finding relationships between areas, angles, and sides.

Answer 1

The Correct Option is C

Step 1: Analyze Statement 1.
Statement 1 tells us that \( QS \) is perpendicular to \( PR \) and has a length of 12. This gives us information about the geometry of the triangle, but we do not yet know the dimensions of the other sides, so statement 1 alone is not sufficient to answer the question.
Step 2: Analyze Statement 2.
Statement 2 tells us that the perimeter of triangle PQR is 60. While this provides the total perimeter of the triangle, it does not give us the individual side lengths or the relationship between the areas of the triangles PQS and QRS, so statement 2 alone is also insufficient.
Step 3: Combine Both Statements.
Combining both statements, we know the length of \( QS \) and the perimeter of triangle PQR. With this information, we can determine the areas of triangles PQS and QRS by using the properties of right-angled triangles and the given perimeter. Therefore, both statements together are sufficient to answer the question.
Step 4: Conclusion.
The correct answer is (C).