Question 711640
In a golden rectangle, length {{{L}}} and width {{{W=x}}} are related by the ratio
{{{L/W=(1+sqrt(5))/2}}} --> {{{L=W(1+sqrt(5))/2}}} --> {{{L=x(1+sqrt(5))/2}}}
The diagonal splits the rectangle into 2 congruent right triangles.
The legs lengths are {{{L}}} and {{{W=x}}},
and the hypotenuse is the diagonal so its length is {{{10cm}}}.
The Pythagorean theorem says
{{{L^2+W^2=10^2}}} --> {{{(x(1+sqrt(5))/2)^2+x^2=100}}} --> {{{x^2(1+sqrt(5))^2/2^2+x^2=100}}} --> {{{x^2(1+2sqrt(5)+5)/4+x^2=100}}}
Multiplying both sides of the equal sign times 4  we get
{{{x^2(1+2sqrt(5)+5)+4x^2=400}}} --> {{{x^2(1+2sqrt(5)+5+4)=400}}} --> {{{x^2(2sqrt(5)+10)=400}}} --> {{{x^2=400/(2sqrt(5)+10)}}}
At this point, we can calculate an approximate value for {{{x^2}}} as a decimal number,
and then find its square root for a good approximation of {{{x}}}.
If we want an exact value, all we can get is a more or less complicated expression with square roots:
{{{x^2=400/(2sqrt(5)+10)}}} --> {{{x^2=200/(sqrt(5)+5)}}} --> {{{x^2=200(5-sqrt(5))/((5-sqrt(5))(5+sqrt(5)))}}}
 --> {{{x^2=200(5-sqrt(5))/(25-5)}}} --> {{{x^2=200(5-sqrt(5))/20}}} --> {{{x^2=10(5-sqrt(5))}}} --> {{{x^2=50-10sqrt(5))}}} --> {{{x=sqrt(50-10sqrt(5)))}}}