Question

How do you do this question?

Answer 1

The length of a circular arc (s) is the product of the circle's radius and the central angle measure in radians.

... s = r·θ

Then the central angle measure in radians is ...

... θ = s/r

For your problem, you have s = 8π/9 and r = 1. Then the central angle AOC is

... ∠AOC = (8π/9)/(1) = 8π/9 . . . . radians

The relationship between radians and degrees is

... 180° = π radians

Multiplying this equation by 8/9 will tell us the degree measure of ∠AOC.

... 8/9×180° = (8/9)×π radians

... ∠AOC = (8/9)·180° = 160°

We know that this is the sum of the two (equal) central angles ∠AOB and ∠BOC, so we have

... 2×∠AOB = 160°

... ∠AOB = 160°/2 = 80°

_____

Here, you know that 8π/9 is the central angle in radians because the ratio of arc length to radius is the angle in radians.

Answer 2

The central angle corresponding to AOB is half of the 8pi/9 radians mentioned, meaning that the central angle of AOB is 4pi/9.

Converting that to degrees:

4pi/9 180 degrees

-------- * -------------------- = 80 degrees.

1 pi