Clock geometry — Between 8 o’clock and 9 o’clock, at what time are the hands of a clock at right angles (90 degrees)?

Introduction / Context:
Right angles occur when the hand-angle difference is 90 degrees. We find solutions between 8:00 and 9:00.

Given Data / Assumptions:

• At 8:00, hour angle = 240°.
• Let m be minutes after 8:00.
• Hour angle = 240 + 0.5m; minute angle = 6m.

Concept / Approach:
Set |(240 + 0.5m) − 6m| = 90 and solve for m within (0, 60).

Step-by-Step Solution:
|240 − 5.5m| = 90 → 5.5m = 240 ± 90 = 330 or 150.m = 330/5.5 = 60 (hits exactly 9:00, boundary) or m = 150/5.5 = 300/11 ≈ 27 3/11.Valid between 8 and 9: 8 : 300/11.

Verification / Alternative check:
Plugging m = 300/11 satisfies the 90° separation.

Why Other Options Are Wrong:
163/11, 273/11, and 173/11 are not equal to 300/11 and do not yield the correct angle within the hour.

Common Pitfalls:
Accepting the boundary time m = 60 as “between 8 and 9”; most questions exclude the hour boundary.

Final Answer:
8 : 300/11 (not listed), hence “None of these.”