If the lengths of sides AB, BC and AC in the figure shown form a geometric progression in that order, what is the ratio between AC and AB to 3 decimal places?
+23254 For this geometric progression, let a be the multiplier.
Let x be the length of AB.
Then a·x is the length of BC,
and a2x is the length of AC.
Since they are the legs of a right triangle: (x)2 + (ax)2 = (a2x)2
x2 + a2x2 = a4x2
dividing by x2: 1 + a2 = a4
rewriting: a4 - a2 - 1 = 0
Using the quadratic formula: a2 = [1 +/- sqrt( 12 + 4 ) ] / 2
ignoring the minus because it is a length: a2 = [1 + sqrt( 5 ) ] / 2
This means that AC = a2x = { [1 + sqrt( 5 ) ] / 2 } · x
and AB = x
Dividing AC by AB: { [1 + sqrt( 5 ) ] / 2 } · x / x = [1 + sqrt( 5 ) ] / 2
