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The length of LN is 65.

Step-by-step explanation:

In MNK,

h² = p² + b²

or, 90² = 54²+b²

or, 8100 = 2916+b²

or, b² = 5184

so, b = 72

KN = 72

Now,

In LKN,

h²=p²+b²

or, 97²=LN²+KN²

or, 9409 = LN² + 72²

or, 9409 = LN² + 5184

or, LN² = 4225

so, LN = 65

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.

a

step-by-step explanation: