Question

The hands of clock form an acute angle at $2$ O’ clock. What type of angle do they form at $4$ O’ clock?
A. Acute
B. Right
C. Obtuse
D. Straight

Answer 1

Hint: We have the formula for the angle between the hands of the clock for a given time as $\theta =|30H-5.5M|$. From this formula, we will calculate the angle formed between the clock hands when time is $2$ O’ clock. But in the problem, we have that the angle formed between the clock hands when time is $2$ O’clock is an acute angle. Now we will calculate the angle between the clock hands when the time is $4$ O’clock. For the obtained value we will check whether the obtained angle is Acute or Right or Obtuse or Straight or reflex or full rotation.

Complete step by step answer:
Given that,
Hands of the clock make an acute angle when time is $2$ O’clock.
Now the angle made by the hands of the clock is given by
 $\begin{array}{rl}& \theta =|30H-5.5M|\\ & \Rightarrow \theta =|30\times 2-5.5\times 0|\\ & \Rightarrow \theta =60{}^{\circ }\end{array}$
We know that the angle $60{}^{\circ }$ is less than $90{}^{\circ }$ , so the angle $60{}^{\circ }$ an acute angle.
Now the angle made by the hands of the clock when time is $4$ O’ clock is given by
 $\begin{array}{rl}& \theta =|30H-5.5M|\\ & \Rightarrow \theta =|30\times 4-5.5\times 0|\\ & \Rightarrow \theta =120{}^{\circ }\end{array}$
Here the angle $120{}^{\circ }$ is greater than $90{}^{\circ }$ , so the angle $120{}^{\circ }$ is known as Obtuse angle. It can be clearly shown in below figure

Hence option – C is the correct answer.

Note:
 While dealing with this kind of problem it is better to know about the classifications of the angles. Angles are divided into six types namely Acute Angle, Right Angle, Obtuse Angle, Straight Angle, Reflex Angle, Full Rotation. We can also note that the angle would be the right angle when it is 3 O’clock and 9 O’clock and it would be a straight angle when it is 6 O’clock.