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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?

May 11, 2020

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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

May 11, 2020