In the diagram, DC bisects

We are given that ∠BDA is bisected

this means that ∠BDC = ∠ADC

Finding x :

Since ∠BDC = ∠ADC:

7x - 19 = 9x - 37

2x = 18

x = 9

Finding ∠ADC:

We are given that ∠ADC = 9x - 37

now that we know that x = 9, replacing x:

∠ADC = 9(9)-  37

∠ADC = 44°

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

yes. we do proportions in probabilities.