Henry Ernest Dudeney/Puzzles and Curious Problems/62 - At Right Angles/Solution/Proof 1

by : $62$

 * At Right Angles

Proof
Let $T$ be the time in question.

Let $m$ be the number of minutes past $5:00$ that $T$ is.

Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock at time $T$.

Let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock at time $T$.

Between $5:00$ and $6:00$ we have that $150 < \phi < 180$.

Hence from Condition for Valid Time Indication, we have that:
 * $12 \paren {\phi - 150} = \theta$

Then we are told that:
 * $\theta = \phi - 90$

as (from how soon) we are interested in the earliest instance of the hands being at right angles.

Hence:

Hence the result.