Question

\( AB \) is a diameter of the circle. Compare: 
Quantity A: The length of \( AB \) 
Quantity B: The average (arithmetic mean) of the lengths of \( AC \) and \( AD \). 

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.
• E.

Hint:

In circle geometry, the diameter is the longest chord. This fact often helps in direct comparisons without calculations.

Answer 1

The Correct Option is A

Step 1: Property of a diameter.
In a circle, the diameter is the longest chord. Hence, \( AB \), being a diameter, is longer than any other chord such as \( AC \) and \( AD \).
Step 2: Comparing \( AB \) with other chords.
Since \( AC \) and \( AD \) are not diameters, we have: \[ AB>AC \quad \text{and} \quad AB>AD. \] Step 3: Arithmetic mean property.
The arithmetic mean of two numbers is always less than or equal to the larger number. Therefore, \[ \frac{AC + AD}{2}<AB. \] Step 4: Conclusion.
Thus, Quantity A (length of \( AB \)) is always greater than Quantity B (average of \( AC \) and \( AD \)). The correct answer is: \[ \boxed{\text{(A) Quantity A is greater.}} \]