Answer: 34 degrees
===========================================================
Explanation:
PC = 12 is the radius, so 2*12 = 24 is the diameter
Circumference = pi*diameter = pi*24 = 24pi
-----------
Let x be the central angle APC.
To find the length of minor arc AC, we use this formula
L = (x/360)*(circumference)
L is the arc length and x is defined as above. Plug in 24pi for the circumference and plug in L = 68pi/15 which is the shortest distance from A to C along the circle. Solve for x
-----------
L = (x/360)*(circumference)
68pi/15 = (x/360)*(24pi)
68pi/15 = pi*x/15
68/15 = x/15 ... divide both sides by pi, the pi terms cancel
68 = x .... multiply both sides by 15, the denominators cancel
x = 68
The central angle APC is 68 degrees. Therefore, minor arc AC is 68 degrees as well.
Now use the inscribed angle theorem. This says that the inscribed angle is half that of the arc it cuts off. This means inscribed angle ABC is half that of minor arc AC.
angle ABC = (1/2)*(minor arc AC)
angle ABC = (1/2)*(68)
angle ABC = 34 degrees
