Clock – Straight line (180°) between 8:00 and 9:00: At what time between 8 and 9 o'clock will the hands of a clock be in the same straight line but not together?

Introduction / Context:
“Same straight line but not together” means the angle is 180° (opposite directions). We seek the time between 8:00 and 9:00 satisfying this geometry condition.

Given Data / Assumptions:

• Hour-hand speed: 0.5°/min.
• Minute-hand speed: 6°/min.
• At 8:00, hour hand at 240°.

Concept / Approach:
Angle formula between hands at h:m is |30h − 5.5m|. For h=8, set |240 − 5.5m| = 180 and solve for m within (0, 60).

Step-by-Step Solution:
240 − 5.5m = ±180.Case +: 240 − 5.5m = 180 → 5.5m = 60 → m = 10 10/11.Case −: 240 − 5.5m = −180 → 5.5m = 420 → m = 76 4/11 (not in the hour window).

Verification / Alternative check:
Plug m = 10 10/11 back to confirm 180°.

Why Other Options Are Wrong:
Times tied to 5 o’clock or a second 8-o’clock solution beyond 60 minutes are invalid.

Common Pitfalls:
Expecting two solutions for every hour; sometimes only one lies in the interval.

Final Answer:
10 10/11 minutes past 8