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Right Angle Between Hands

Introduction

This type of question asks: “At what time are the hour and minute hands of a clock exactly at a right angle (90°)?” Learning this helps you handle not just 90° cases but also angles like 45°, 135°, and 180° using the same logic.

Pattern: Right Angle Between Hands

Pattern

The key idea is to find when the hands are 90° apart using this formula:
|(30 × Hour) - (11/2 × Minutes)| = 90

This equation gives two possible results because the hands can form a right angle in two directions (one before and one after overlapping):

  • When the minute hand is ahead of the hour hand: m = (60/11)(H + 3)
  • When the minute hand is behind the hour hand: m = (60/11)(H - 3)

Only values of m between 0 and 60 are valid (minutes past that hour).

Step-by-Step Example

Question

Between 2 and 3 o’clock, at what time are the hands of a clock at a right angle?

Solution

  1. Step 1: Write the formula

    |(30 × Hour) - (11/2 × Minutes)| = 90

  2. Step 2: Substitute H = 2

    |(30 × 2) - (11/2 × m)| = 90 → |60 - 5.5m| = 90

  3. Step 3: Remove the absolute value

    We’ll get two equations:
    (a) 60 - 5.5m = 90 and (b) 60 - 5.5m = -90

  4. Step 4: Solve both equations

    From (a): 5.5m = -30 → m = -5.45 (not possible, as minutes can’t be negative).
    From (b): 5.5m = 150 → m = 27.27 minutes = 27 3/11 minutes.

  5. Step 5: Interpret the result

    Only the positive value makes sense in the 2-3 interval. Therefore, the right angle occurs at 2:27 3/11.

  6. Step 6: Verify next interval

    The next right angle happens in the following hour (3-4) at 3:32 8/11.

  7. Final Answer:

    Between 2 and 3 → 2:27 3/11
    Between 3 and 4 → 3:00 and 3:32 8/11

  8. Quick Check:

    Substitute 2:27 3/11 → (30×2) - (11/2×27.27) = 60 - 150 = -90 → |-90| = 90° ✅

Quick Variations

1. You can use the same method for any hour by replacing H in the formula.

2. If you get a negative minute, it means the right angle occurred in the previous hour.

3. For other angles (like 45° or 135°), replace 90 with that value in the same formula.

Trick to Always Use

  • Use the simple formula: |30H - 11M/2| = 90.
  • Always check for two times - one before and one after overlap.
  • Ignore negative or more than 60-minute values; they belong to adjacent hours.

Summary

Summary

  • Right angles occur when |30H - 11M/2| = 90.
  • For each hour, there are usually two such times (except near 2-4 and 8-10).
  • Valid minute values must be between 0 and 60.
  • If you get a negative or >60 result, it means the angle occurs outside that hour.

Example to remember:
Between 2 and 3 → 2:27 3/11
Between 3 and 4 → 3:00 and 3:32 8/11

Practice

(1/5)
1. At what time between 1 and 2 o’clock will the hands of a clock be at a right angle for the first time?
easy
A. 1:21 9/11
B. 1:27 3/11
C. 1:32 8/11
D. 1:15 5/11

Solution

  1. Step 1: Write the right-angle condition

    |(30 × H) - (11/2 × m)| = 90.

  2. Step 2: Substitute H = 1

    |30 - 5.5m| = 90.

  3. Step 3: Remove the absolute value and solve

    Case (a): 30 - 5.5m = 90 → 5.5m = -60 → m = -60/5.5 (invalid).
    Case (b): 30 - 5.5m = -90 → 5.5m = 120 → m = 120/5.5 = 21 9/11 minutes.

  4. Step 4: Interpret

    m = 21 9/11 is within 0-60, so the first right angle is at 1:21 9/11.

  5. Final Answer:

    1:21 9/11 → Option A

  6. Quick Check:

    Hour position = 30×1 + 0.5×(21 9/11) ≈ 30 + 10.545 = 40.545°; minute = 6×21.818 ≈ 130.909°; difference ≈ 90° ✅
Hint: Use |30H - 11M/2| = 90 and discard negative minute roots for that hour.
Common Mistakes: Accepting negative minute values as valid for the same hour.
2. At what time between 4 and 5 o’clock will the hands of a clock be at right angles for the first time?
easy
A. 4:05 5/11
B. 4:16 4/11
C. 4:27 3/11
D. 4:38 2/11

Solution

  1. Step 1: Use the general formulas

    m = (60/11)(H ± 3).

  2. Step 2: Compute for H = 4

    m₁ = (60/11)(4 - 3) = 60/11 = 5 5/11 minutes.
    m₂ = (60/11)(4 + 3) = 420/11 = 38 2/11 minutes.

  3. Step 3: Choose the first occurrence after 4:00

    The earlier valid time is 4:05 5/11 (m₁).

  4. Final Answer:

    4:05 5/11 → Option A

  5. Quick Check:

    Hour position = 30×4 + 0.5×5.454 ≈ 120 + 2.727 = 122.727°; minute = 6×5.454 ≈ 32.727°; difference ≈ 90° ✅
Hint: Compute (60/11)(H - 3) for the earlier right angle in the hour.
Common Mistakes: Mixing up which of the ± roots is earlier in the hour.
3. At what time after 5:00 (between 5 and 6) will the hands of a clock be at a right angle for the first time?
easy
A. 5:10 10/11
B. 5:21 9/11
C. 5:16 4/11
D. 5:32 8/11

Solution

  1. Step 1: Use m = (60/11)(H - 3) or (60/11)(H + 3)

  2. Step 2: Substitute H = 5

    m₁ = (60/11)(5 - 3) = (60/11)×2 = 120/11 = 10 10/11 minutes.
    m₂ = (60/11)(5 + 3) = (60/11)×8 = 480/11 = 43 7/11 minutes.

  3. Step 3: The first right angle after 5:00

    is at 5:10 10/11 (m₁).

  4. Final Answer:

    5:10 10/11 → Option A

  5. Quick Check:

    Hour = 30×5 + 0.5×10.909 ≈ 150 + 5.455 = 155.455°; minute ≈ 65.455°; difference ≈ 90° ✅
Hint: For the earlier time use (60/11)(H - 3) when it stays positive.
Common Mistakes: Rounding fractions too early and losing precision.
4. How many times do the hands of a clock make a right angle between 12 noon and 6 p.m.?
medium
A. 10
B. 11
C. 12
D. 13

Solution

  1. Step 1: Understand the hourly pattern

    Typically, hands form a right angle twice in an hour (one when minute hand is ahead, one when behind), giving 2 per hour.

  2. Step 2: Count hours from 12 to 6

    There are 6 hours (12→1, 1→2, …, 5→6). If each hour gives 2 right angles, total = 6 × 2 = 12.

  3. Step 3: Check exceptions

    Special-hour exceptions (where a root falls outside the hour) do not remove a count within 12-6 range, so total remains 12.

  4. Final Answer:

    12 → Option C

  5. Quick Check:

    2 right angles × 6 hours = 12 ✅
Hint: Multiply the number of hours by 2 (for standard intervals).
Common Mistakes: Forgetting that some hours may shift a root into adjacent hour-verify when needed.
5. At what time between 8 and 9 o’clock will the hands of a clock be at a right angle?
medium
A. 8:05 5/11
B. 8:16 4/11
C. 8:27 3/11
D. 8:49 1/11

Solution

  1. Step 1: Set H = 8 in m = (60/11)(H ± 3)

  2. Step 2: Compute both roots

    m₁ = (60/11)(8 - 3) = (60/11)×5 = 300/11 = 27 3/11 minutes.
    m₂ = (60/11)(8 + 3) = (60/11)×11 = 60 → corresponds to 9:00 (not inside 8-9).

  3. Step 3: Choose the valid time in 8-9

    The valid right angle in 8-9 is 8:27 3/11.

  4. Final Answer:

    8:27 3/11 → Option C

  5. Quick Check:

    Hour ≈ 30×8 + 0.5×27.273 = 240 + 13.636 = 253.636°; minute = 6×27.273 = 163.636°; difference = 90° ✅
Hint: If (60/11)(H + 3) = 60, it means the other right angle is exactly at the next hour (e.g., 9:00).
Common Mistakes: Assuming both ± roots always lie within the same hour.

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