Question 1030322
First we multiply them: 12×1/27 = 12/27 = 0.4444
Then take the square root: √0.4444 = 0.66667 This is your geometric mean
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TOA: Tangent(theta) = opposite/adjacent
5.5 ft tall - standing 54 ft from tree 
47 degrees
tan 47 = x/54
1.0724 = x/54
x = 1.0724*54
x = 57.9 To this add her height
. . 5.5
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. . 63.4 feet is the height of the tree
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SOH: Sine (theta) = opposite/hypotenuse
sin (67)= 121/h
0.9205 = 121/h multiply both sides by 121
h = 0.9205*121 = 111.38 is the hypotenuse of your triangle which in this case is the length of the string assuming the string was perfectly straight.
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I don't understand your last problem.