Question

Which has the greater length, AB or CD?

Statement 1: CD is the diameter of the circle.
Statement 2: AB is the length of the side of a square inscribed in a circle whose radius is half of CD.

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Work out AB in terms of CD using the inscribed square's diagonal to side relationship.

Answer 1

The Correct Option is B

Step 1: Read what each statement offers.
The question asks us to compare two lengths, AB and CD.

Step 2: Check statement 1 alone.
Statement 1 only says CD is the diameter of some circle.
It gives no information about AB at all, so no comparison is possible.
Statement 1 alone is not sufficient.

Step 3: Check statement 2 alone.
Statement 2 says AB is the side of a square inscribed in a circle whose radius equals half of CD, that is, radius \( = \frac{CD}{2} \).
The diagonal of a square inscribed in a circle equals the circle's diameter, so this square's diagonal is \( 2 \times \frac{CD}{2} = CD \).
For a square, side \( = \frac{\text{diagonal}}{\sqrt{2}} \), so \( AB = \frac{CD}{\sqrt{2}} \).
Since \( \frac{1}{\sqrt{2}} \approx 0.71 \), this makes AB smaller than CD, so CD is definitely the greater length.
Statement 2 alone answers the question fully.

Step 4: Confirm sufficiency.
Statement 2 links AB directly to CD through the geometry of the inscribed square, without needing statement 1 at all.

Final Answer:
Statement 2 alone is sufficient, and CD is the greater length. \[ \boxed{\text{Statement (2) alone is sufficient (option b)}} \]