Question

What is the area of the circle shown above with center O?
I. W is the mid-point of chord XY.
II. The ratio of ZW to OW is 3:5

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II 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 I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In geometry-based Data Sufficiency questions, be wary of statements that only provide ratios or proportionalities. To find a concrete area or length, you almost always need at least one statement that provides a specific numerical measurement (e.g., a length, an angle in degrees, or an area).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a data sufficiency question. To find the area of the circle, we need to determine its radius, \(r\). The area is given by the formula \(A = \pi r^2\). We must evaluate if the given statements, alone or together, allow us to find a unique numerical value for \(r\). 
Step 2: Detailed Explanation: 
Analyze Statement I: "W is the mid-point of chord XY." 
This statement tells us a geometric property. When a line from the center of a circle bisects a chord, it is perpendicular to the chord. Therefore, \(\angle OWY = 90^\circ\). This forms a right-angled triangle \(\triangle OWY\), where OY is the radius (\(r\)). According to the Pythagorean theorem, \(OY^2 = OW^2 + WY^2\), or \(r^2 = OW^2 + WY^2\). This statement gives us a relationship between lengths but provides no numerical values. Thus, Statement I alone is not sufficient. 
Analyze Statement II: "The ratio of ZW to OW is 3:5." 
Let the radius be \(r\). OZ is a radius, so \(OZ = r\). From the diagram, we can see that \(OZ = OW + WZ\). The statement gives us \(\frac{ZW}{OW} = \frac{3}{5}\), or \(ZW = \frac{3}{5}OW\). 
Substituting this into the radius equation: \(r = OW + \frac{3}{5}OW = \frac{8}{5}OW\). 
This gives us a relationship between the radius \(r\) and the length of the segment OW, but no actual numerical values. Thus, Statement II alone is not sufficient. 
Analyze Statements I and II Together: 
From Statement I, we have \(r^2 = OW^2 + WY^2\). 
From Statement II, we have \(r = \frac{8}{5}OW\), which means \(OW = \frac{5}{8}r\). 
Substituting the expression for OW into the first equation: 
\[ r^2 = \left(\frac{5}{8}r\right)^2 + WY^2 \] 
\[ r^2 = \frac{25}{64}r^2 + WY^2 \] 
\[ WY^2 = r^2 - \frac{25}{64}r^2 = \frac{39}{64}r^2 \] 
This gives us the length of WY in terms of \(r\), but we still do not have a numerical value for \(r\) or any other length. Without any specific length measurement, we cannot calculate the area. Therefore, both statements together are not sufficient. 
Step 3: Final Answer: 
Since even with both statements combined, we cannot determine a specific value for the radius, additional data is needed. This corresponds to option (E).