Question:medium

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.

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Work out AB in terms of CD using the inscribed square's diagonal to side relationship.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Frame the comparison with a formula.
For a square inscribed in a circle of radius r, its side works out to $ r\sqrt{2} $, since the diagonal is $ 2r $ and side $ = \frac{diagonal}{\sqrt{2}} = r\sqrt{2} $.

Step 2: Test statement 1 by itself.
Statement 1 tells us CD is a diameter, but says nothing that connects to AB.
No comparison can follow from this fact alone.

Step 3: Test statement 2 by itself.
Statement 2 fixes the circle's radius as $ \frac{CD}{2} $, so using the formula from Step 1, $ AB = \frac{CD}{2} \times \sqrt{2} = \frac{CD}{\sqrt{2}} $.
Numerically $ \frac{1}{\sqrt{2}} $ is close to 0.71, well under 1, so AB comes out smaller than CD every time.
This statement alone settles the comparison without touching statement 1.

Step 4: Rule out needing statement 1.
Since statement 2 already ties AB to CD through the square's geometry, adding statement 1 changes nothing.

Final Answer:
Statement 2 by itself proves CD is longer than AB. \[ \boxed{\text{Statement (2) alone is sufficient (option b)}} \]
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