Discussion

At what time after 9 am, do hands of clock make a right angle?

Answer: A

The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is

\[ \theta = \left| 30H - 5.5M \right| \]

A right angle means \( \theta = 90 \), with \( H = 9 \):

\[ \left| 270 - 5.5M \right| = 90 \]

\[ 5.5M = 180 \quad\text{or}\quad 5.5M = 360 \]

\[ M = \frac{360}{11} = 32\tfrac{8}{11} \quad\text{or}\quad M = 65\tfrac{5}{11} \text{ (past the hour)} \]

So the first right angle after 9 am comes at \( 32\tfrac{8}{11} \) minutes past 9.

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