Start with calculating the measure of the angle ∠ AOM, applying the concept of the cosine to △ AOM.
mAB=120^(∘)
Practice makes perfect
We are given the following diagram and asked to find the measure of the arc AB.
It is given that the radius of the circle measures 12 units. Segment AO is also a radius, so its length is also 12.
Now, let's consider a right triangle △ AOM. We know the lengths of its two sides, so we can find the measure of ∠ AOM using the concept of cosine.
cos(m∠ AOM)=OM/AO
Let's substitute OM with 6 and AO with 12 and find the value of m∠ AOM.
The cosine of the angle is 12, which indicates that the angle is 60^(∘).
m∠ AOM=60^(∘)
Next, we will draw a segment OB and consider the triangle △ AOB.
As we can see, OB is also a radius of the circle, so its length is also 12. This implies that △ AOM is an isosceles triangle. OM is a median of AB in the isosceles triangle, so it is also a bisector of ∠ AOB.
Therefore, m∠ BOM=60^(∘). Adding the measures of ∠ AOM and ∠ BOM, we can calculate the measure of the central angle ∠ AOB.
m∠ AOB=60^(∘)+60^(∘)=120^(∘)
Finally, let's notice that ∠ AOB intercepts the arc AB. The measure of an arc is equal to the measure of its central angle, which allows us to conclude that mAB=120^(∘).
