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 \( AD \).
\( AB \) 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 \( AD \).
Show Hint
In circle geometry, the diameter is the longest chord. This fact often helps in direct comparisons without calculations.
Updated On: Oct 7, 2025
- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.
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The Correct Option is A
Solution and Explanation
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: \[ \boxed{\text{(A) Quantity A is greater.}} \]
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: \[ \boxed{\text{(A) Quantity A is greater.}} \]
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