Question

What is the area of the circle shown above with center O? 

(1) W is the mid-point of chord XY. 
(2) The ratio of ZW to OW is 3:5. 
 

• 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:

When dealing with geometry problems, always check if the given statements provide enough information about the radius or other necessary measurements to solve the problem.

Answer 1

The Correct Option is C

Step 1: Analyze Statement 1.
Statement 1 tells us that W is the mid-point of chord XY. While this gives some information about the geometry of the circle, it does not provide enough information to directly find the area of the circle. The radius or any other numerical information is still missing. Therefore, statement 1 alone is not sufficient.
Step 2: Analyze Statement 2.
Statement 2 tells us that the ratio of ZW to OW is 3:5. This gives some information about the relative lengths of two segments, but it alone does not provide enough information to determine the area of the circle. The exact lengths or the radius are not given, so statement 2 alone is also insufficient.
Step 3: Combine Both Statements.
By combining both statements, we know that W is the mid-point of chord XY, and the ratio of ZW to OW is 3:5. This information, combined with the properties of the circle and the right triangle formed by ZW and OW, can allow us to solve for the radius of the circle. Once we know the radius, we can easily calculate the area of the circle. Thus, both statements together are sufficient.
Step 4: Conclusion.
The correct answer is (C), as both statements together are sufficient to find the area of the circle but neither statement alone is sufficient.