What is the first time after 1 o’clock that the hands of a clock will be 90°

To find the first time after 1 o'clock when the hands of a clock will form a 90° angle, we need to consider the positions of the hour and minute hands.

### Step-by-Step Solution:

1. **Hour Hand Movement:**
   The hour hand moves 30° per hour (since there are 360° total and 12 hours). After 1 o'clock, at any time \( t \) minutes after 1 o'clock, the hour hand will have moved:
   \[
   \text{Hour hand angle} = 30^\circ + \frac{1}{2}t
   \]
   where \( t \) is in minutes. This is because the hour hand moves 0.5° per minute.

2. **Minute Hand Movement:**
   The minute hand moves 6° per minute (since it moves 360° in 60 minutes). So at \( t \) minutes after 1 o'clock, the minute hand will be at:
   \[
   \text{Minute hand angle} = 6t
   \]

3. **Condition for 90° Angle:**
   We want the absolute difference between the angles of the hour and minute hands to be 90°. Therefore, we set up the equation:
   \[
   |(30 + 0.5t) - 6t| = 90
   \]

   This gives us two possible cases:
   
   **Case 1:**
   \[
   (30 + 0.5t) - 6t = 90
   \]
   Simplifying:
   \[
   30 - 5.5t = 90
   \]
   \[
   -5.5t = 60
   \]
   \[
   t = \frac{60}{5.5} \approx 10.91 \text{ minutes}
   \]

   **Case 2:**
   \[
   6t - (30 + 0.5t) = 90
   \]
   Simplifying:
   \[
   5.5t - 30 = 90
   \]
   \[
   5.5t = 120
   \]
   \[
   t = \frac{120}{5.5} \approx 21.82 \text{ minutes}
   \]

### Answer:
The first time after 1 o'clock that the hands of the clock form a 90° angle is approximately **10.91 minutes**, which is around **1:11 AM**.

Would you like more details on this, or do you have any questions?

### Related Questions:
1. When will the hands of the clock form a straight line after 1 o'clock?
2. How many times between 1 o'clock and 2 o'clock do the hands of a clock form a 90° angle?
3. What is the next time after 1:11 that the hands of a clock form a 90° angle?
4. How long after any hour will the hands of a clock overlap?
5. How often do the hands of a clock form a right angle during a 12-hour period?

### Tip:
The hands of a clock align at a right angle (90°) exactly twice within each hour.

What is the first time after 1 o’clock that the hands of a clock will be 90°?

Discover how to calculate the first time after 1:00 when the clock hands form a 90° angle. This step-by-step solution covers key concepts in geometry and clock angle problems. Learn the formulas for the hour and minute hand angles, with calculations suitable for students in Grades 8-10.

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