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 square inscribed in a circle whose radius is half of CD.

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is B

To determine which has the greater length between AB and CD, let's analyze the given statements:

1. Statement 1: CD is the diameter of the circle.
   • This statement tells us that CD is the diameter of the circle. However, without additional information about AB, such as measurements or a relationship with CD, we cannot determine which is longer.
2. Statement 2: AB is the length of the side of a square inscribed in a circle whose radius is half of CD.
   • Let's further analyze this statement:
   • If CD is the diameter of the circle, then the radius of this circle is \(R = \frac{CD}{2}\).
   • The radius of the circle in which the square is inscribed is half of CD, so its radius is \(r = \frac{CD}{4}\).
   • In a circle, the diagonal of the inscribed square is equal to the diameter of the circle. Hence, the diagonal of the square is \(2r = \frac{CD}{2}\).
   • For a square, if \(s\) is the side length, then the relation between side and diagonal \(d\) is given by \(d = s\sqrt{2}\).
   • Therefore, we have: \(\frac{CD}{2} = s\sqrt{2}\).
   • Solving for \(s\), we get: \(s = \frac{CD}{2\sqrt{2}}\) or \(s = \frac{CD\sqrt{2}}{4}\).
3. Comparing \(s\) and \(CD\):
   • We can clearly see that \(s \lt CD\), because \(CD \gt \frac{CD\sqrt{2}}{4}\).
   • Hence, AB (side of the square) is shorter than CD (diameter of the circle).

Thus, Statement 2 alone is sufficient to conclude that CD has greater length than AB.