SOLUTION: I have a golden rectangle with a diagonal of 10 cm. What is it width value [x]? No length value is given. How do I solve for x?

Algebra ->  Rectangles -> SOLUTION: I have a golden rectangle with a diagonal of 10 cm. What is it width value [x]? No length value is given. How do I solve for x?       Log On


   

Question 711640: I have a golden rectangle with a diagonal of 10 cm. What is it width value [x]? No length value is given.
How do I solve for x?

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
In a golden rectangle, length L and width W=x are related by the ratio
L%2FW=%281%2Bsqrt%285%29%29%2F2 --> L=W%281%2Bsqrt%285%29%29%2F2 --> L=x%281%2Bsqrt%285%29%29%2F2
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%5E2%2BW%5E2=10%5E2 --> %28x%281%2Bsqrt%285%29%29%2F2%29%5E2%2Bx%5E2=100 --> x%5E2%281%2Bsqrt%285%29%29%5E2%2F2%5E2%2Bx%5E2=100 --> x%5E2%281%2B2sqrt%285%29%2B5%29%2F4%2Bx%5E2=100
Multiplying both sides of the equal sign times 4 we get
x%5E2%281%2B2sqrt%285%29%2B5%29%2B4x%5E2=400 --> x%5E2%281%2B2sqrt%285%29%2B5%2B4%29=400 --> x%5E2%282sqrt%285%29%2B10%29=400 --> x%5E2=400%2F%282sqrt%285%29%2B10%29
At this point, we can calculate an approximate value for x%5E2 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%5E2=400%2F%282sqrt%285%29%2B10%29 --> x%5E2=200%2F%28sqrt%285%29%2B5%29 --> x%5E2=200%285-sqrt%285%29%29%2F%28%285-sqrt%285%29%29%285%2Bsqrt%285%29%29%29
--> x%5E2=200%285-sqrt%285%29%29%2F%2825-5%29 --> x%5E2=200%285-sqrt%285%29%29%2F20 --> x%5E2=10%285-sqrt%285%29%29 --> x%5E2=50-10sqrt%285%29%29 --> x=sqrt%2850-10sqrt%285%29%29%29