a Let's consider a circular clock. The clock is divided into 12 congruent sections. We are asked to determine the measure of each arc in this circle.
Dividing the clock into 12 congruent sections is equivalent to dividing the 360^(∘) angle into 12 congruent angles.
360^(∘)/12 = 30^(∘)
This means that the central angle measure of each segment is 30^(∘). Since by definition the measure of an arc is the measure of its central angle, the measure of each arc is 30 ^(∘).
b Let's position the hour and minute hands so that it shows 7:00 o'clock.
As we can see, the large arc formed by the hands of the clock is made up of 5 smaller arcs represented by the hour segments. Since these small arcs measure 30^(∘), the measure of the arc we are asked to find is 5*30^(∘)=150^(∘).
5 * 30^(∘) = 150^(∘)
c Now we want to form an arc that is congruent to the arc in Part B. In other words, we want to create an arc whose measure is 150^(∘). Let's assume that the time is 12:00.
Now, without changing the position of the minute hand, let's rotate the hour hand so that it covers 5 small arcs, each measuring 30^(∘).
It is 5:00 o'clock! Therefore, the minor arc formed when it is 5:00 is congruent to the minor arc formed when it is 7:00. Please note that there are multiple correct answers; this is just an example.
