Clock – Right angles between 5 and 6: At what times between 5:00 and 6:00 will the hour and minute hands be at right angles (90°)?

Introduction / Context:
Right-angle (90°) problems require setting the angle between the hour and minute hands to 90°. Between 5:00 and 6:00 there are generally two instants where the condition holds.

Given Data / Assumptions:

• Time window: 5:00 to 6:00.
• Minute-hand speed: 6° per minute.
• Hour-hand speed: 0.5° per minute; position at 5:00 is 150°.

Concept / Approach:
Angle formula: |30h − 5.5m| where h is the hour and m is minutes past the hour. For h=5, solve |150 − 5.5m| = 90.

Step-by-Step Solution:
1) Case A: 150 − 5.5m = 90 → 5.5m = 60 → m = 10 10/11 minutes.2) Case B: 150 − 5.5m = −90 → 5.5m = 240 → m = 43 7/11 minutes.3) Both moments lie within (5:00, 6:00).

Verification / Alternative check:
Plug m back into angle formula to confirm 90° (within fractional precision).

Why Other Options Are Wrong:
Options involving times past 6:00 or mis-stated fractions do not satisfy the 5:00–6:00 window or the 90° condition.

Common Pitfalls:
Using 6°−0.5° = 5.5° difference correctly but mixing signs; forgetting there are two solutions.

Final Answer:
10 10/11 min and 43 7/11 min past 5