What would be the type of angle (acute/right/obtuse/straight) between the hour and minute hand when the clock reads the following times? For example, at 10:26, the angle is obtuse.
(Two angles will form between the hour hand and the minute hand. We are talking here only about the smaller angle between the two)
(i) 10:00
(ii) 03:00
(iii) 12:20
(iv) 06:00
(v) 04:40
(vi) 05:25
(vii) 02:30
(viii) 01:40
(ix) 09:00
(x) 11:30
Answer (2)
The types of angles between the hands of the clock at the given times are as follows: 10:00 is acute, 03:00 is right, 12:20 is obtuse, 06:00 is straight, 04:40 is obtuse, 05:25 is acute, 02:30 is obtuse, 01:40 is obtuse, 09:00 is right, and 11:30 is obtuse.
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To determine the type of angle formed between the hour and minute hands of a clock at different times, we need to calculate the smaller angle between them at each given time.
The formula to find the angle between the hour and minute hands is:
θ = ∣ 30 h − 5.5 m ∣
where h is the hour, and m is the minutes past the hour. We always consider the smaller angle, which should be less than 18 0 ∘ .
Here are the angles at the specified times:
(i) 10:00 :
Angle = ∣30 × 10 − 5.5 × 0∣ = ∣300∣ = 6 0 ∘
Type: Acute
(ii) 03:00 :
Angle = ∣30 × 3 − 5.5 × 0∣ = ∣90∣ = 9 0 ∘
Type: Right
(iii) 12:20 :
Angle = ∣30 × 12 − 5.5 × 20∣ = ∣360 − 110∣ = 250 − 180 = 7 0 ∘
Type: Acute
(iv) 06:00 :
Angle = ∣30 × 6 − 5.5 × 0∣ = ∣180∣ = 18 0 ∘
Type: Straight
(v) 04:40 :
Angle = ∣30 × 4 − 5.5 × 40∣ = ∣120 − 220∣ = ∣100∣ = 10 0 ∘
Type: Obtuse
(vi) 05:25 :
Angle = ∣30 × 5 − 5.5 × 25∣ = ∣150 − 137.5∣ = 12. 5 ∘
Type: Acute
(vii) 02:30 :
Angle = ∣30 × 2 − 5.5 × 30∣ = ∣60 − 165∣ = 10 5 ∘
Type: Obtuse
(viii) 01:40 :
Angle = ∣30 × 1 − 5.5 × 40∣ = ∣30 − 220∣ = 190 − 180 = 1 0 ∘
Type: Acute
(ix) 09:00 :
Angle = ∣30 × 9 − 5.5 × 0∣ = ∣270∣ = 9 0 ∘
Type: Right
(x) 11:30 :
Angle = ∣30 × 11 − 5.5 × 30∣ = ∣330 − 165∣ = 16 5 ∘
Type: Obtuse
By following this method, you can easily determine the type and measure of angles at any given hour and minute on the clock.
