If the lengths of sides AB, BC and AC in the figure shown form a geometric progression in that order, what is the ratio AC/AB?
+23246 Let the common ratio be r.
AB = x
BC = x·r
AC = x·r2
AC / AB = x·r2 / x = r2
To find r: the Pythagorean Theorem: ( x )2 + ( x·r )2 = ( x·r2 )2
x2 + x2·r2 = x2·r4
0 = x2·r4 - x2·r2 - x2
0 = x2·( r4 - r2 - 1 )
Divide both sides by x2: 0 = r4 - r2 - 1
Using the quadratic formula: r2 = [ 1 + sqrt( 1 + 4 ) ] / 2
r2 = ( 1 + sqrt(5) ) / 2
AC / AB = r2 = ( 1 + sqrt(5) ) / 2 (phi)
