At what times between 6 and 7 o’clock are the hands of a clock at right angles?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 6:16\(\frac{4}{11}\) and 6:49\(\frac{1}{11}\)
The minute hand gains 5.5° a minute on the hour hand (6° against 0.5°). Between 6 and 7 the hands are at right angles twice: once when the minute hand is 90° behind the hour hand and once when it is 90° ahead.
At 6:00 the hour hand is at \(6\times30^\circ=180^\circ\) and the minute hand at 0°: the minute hand is 180° behind
90° behind: it must gain \(180^\circ-90^\circ=90^\circ\), which takes \(\displaystyle\frac{90}{5.5}=\frac{180}{11}=16\frac{4}{11}\) minutes
90° ahead: it must gain \(180^\circ+90^\circ=270^\circ\), which takes \(\displaystyle\frac{270}{5.5}=\frac{540}{11}=49\frac{1}{11}\) minutes
So the times are 6:16\(\frac{4}{11}\) and 6:49\(\frac{1}{11}\)
