At what time between 5.30 and 6 will the hands of a clock be at right angles?

### Concept & Right Angles A right angle between the clock hands means $\theta = 90^\circ$. This happens twice every hour. Since we are looking for a time between 5:30 and 6:00, the minute hand must be ahead of the hour hand. $$ \theta = |30H - 5.5M| $$ ### Step-by-Step Solution * Given: $H = 5$ and $\theta = 90^\circ$. We need $M > 30$. * Apply formula: $90 = |30(5) - 5.5M| \Rightarrow 90 = |150 - 5.5M|$ * Set up the correct equation for when the minute hand is ahead: $5.5M - 150 = 90$ * Add 150 to both sides: $5.5M = 240$ * Use fraction for 5.5: $\frac{11}{2}M = 240$ * Multiply and divide: $M = \frac{480}{11}$ * Convert to mixed fraction: $M = 43 \frac{7}{11}$ minutes. ### Exam Strategy & Shortcut For right angles, the minute spaces between hands must be 15. At 5 o'clock, the hour hand is at 25 minute spaces. To be at a right angle after 5:30, add 15 spaces: $25 + 15 = 40$ minute spaces. Multiply by the constant $\frac{12}{11}$: $40 \times \frac{12}{11} = \frac{480}{11} = 43 \frac{7}{11}$. ### Common Pitfall Calculating the first right angle (before 5:30) which happens at $25 - 15 = 10$ minute spaces ($10 \times \frac{12}{11} = 10 \frac{10}{11}$ min past 5), ignoring the specific condition "between 5.30 and 6". ### Final Answer Therefore, the correct answer is **$43 \frac{7}{11}$ min. past 5**.