Clock geometry — Between 9 o’clock and 10 o’clock, at what exact time are the hands of a clock in the same straight line but not together (i.e., opposite at 180 degrees)?

Introduction / Context:
“Straight line but not together” means the hands are opposite (180 degrees apart). We seek the time after 9 o’clock when this occurs.

Given Data / Assumptions:

• At 9:00, hour angle = 270°, minute angle = 0°.
• Let t be minutes after 9:00.
• Hour angle = 270 + 0.5t; minute angle = 6t.

Concept / Approach:
Solve |6t − (270 + 0.5t)| = 180 for t in (0, 60).

Step-by-Step Solution:
5.5t = 450 → t = 900/11 ≈ 81.818 (invalid for this hour).5.5t = 90 → t = 180/11 ≈ 16.364 (valid).Thus time = 9 : 180/11.

Verification / Alternative check:
Substituting t = 180/11 gives a 180° separation exactly.

Why Other Options Are Wrong:
164/11, 154/11, 174/11 minutes do not satisfy the 180° condition.

Common Pitfalls:
Choosing the extraneous solution (>60 min) or rounding fractions.

Final Answer:
9 : 180/11 (not listed), therefore “None of these.”