Question

At 3 o’clock, the angle formed between the two hands of a clock is _ _ _ _ _ _ angle.
A) Right
B) Acute
C) Obtuse
D) Left

Answer 1

Hint:
Here we will use the concept that a clock has a shape of a circle. Using this we will find the angle between each number. Then we will multiply the obtained angle to the given time to get the angle between the two hands. We will then choose the option which represents the obtained angle.

Complete step by step solution:
We have to find the angle between the two hands at 3 o’clock.
As a clock is in a circle shape and a circle has \[{360^ \circ }\] angle.
A clock is numbered from 1 to 12 and the total angle is \[{360^ \circ }\].
So, Angle between each number \[ = \dfrac{{{{360}^ \circ }}}{{12}} = {30^ \circ }\]
Next, we will find the angle between two hands at 3 o’clock.
So, at 3 o’clock the minute hand will be on 12 and the hour hand will be on 3.
$\therefore $ The angle between the two numbers \[ = 3 \times {30^ \circ } = {90^ \circ }\]
So, angle between two hands will be \[ = {90^ \circ }\]
We know that \[{90^ \circ }\] represents the right angle.
So, the angle between the two hands at 3 o’clock is \[{90^ \circ }\].

Hence, option (A) is correct.

Note:
There is a direct formula to find the angle between the two hands which is given as,
\[\Delta \theta = |{0.5^ \circ } \times \left( {\left( {60 \times H} \right) - \left( {11 \times M} \right)} \right)|\], where H is the Hour, M is the Minute.
As we have to find the angle at 3 o’clock, So H is 3 and M is 0.
Substituting \[H = 3\] and \[M = 0\] in formula we get,
\[\Delta \theta = |{0.5^ \circ } \times \left( {\left( {60 \times 3} \right) - \left( {11 \times 0} \right)} \right)|\]
Multiplying the terms, we get
\[\Delta \theta = |{0.5^ \circ } \times \left( {180 - 0} \right)|\]
Again multiplying the terms, we get
So, \[\theta = {90^ \circ }\]\[\]