In a circle, if a diameter is perpendicular to a chord then it bisects the chord and its arc.
To apply this theorem we can draw a diameter perpendicular to chord AB.
According to the theorem CD bisects AB, forming two segments that have lengths of 15 units.
Next, we will find the measure of ∠ AOM using the concept of sine.
sin (m∠ AOM)=Opposite/Hypotenuse
The length of the opposite side of the angle is 15, while the length of the hypotenuse is 17. Let's substitute these values into the formula and find the measure of ∠ AOM.
sin (m∠ AOM)=15/17
In order to isolate m∠ AOM, we will use the inverse function of sin.
sin (m∠ AOM)=15/17 ⇔ m∠ AOM=sin ^(- 1)15/17
Finally, let's use a calculator to find the value of the inverse trigonometric ratio. Be sure to check that your calculator is set into degree mode!
The angle is about 62^(∘). Next, we can draw a radius OB and form an isosceles triangle △ AOB, so median and height OM is also a bisector of ∠ AOB.
Thus, the measure of ∠ AOB is twice the measure of ∠ AOM. Let's use this piece of information to calculate m∠ AOB.
m∠ AOB=2m∠ AOM=2(62)=124
Finally, using the fact that the measure of an arc is the same as the measure of its central angle, we conclude that AB — whose central angle is ∠ AOB — measures 124^(∘).
