SOLUTION: A rectangle given is a golden rectangle. ABEF is a square. Rectangle BCDE is similar to rectangle ACDF. a. Show that (a/1) = 1/(a - 1) b. Find the exact value of

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Question 1208158: A rectangle given is a golden rectangle.
ABEF is a square.
Rectangle BCDE is similar to rectangle ACDF.
a. Show that
(a/1) = 1/(a - 1)

b. Find the exact value of a(which will give you the golden ratio) by completing the square.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

We can't tell which rectangle is given as a golden rectangle. Plus, you
didn't tell us what " a ", and what, if anything, you are letting be 1.
Probably English is not your first language.  That is OK!! It is my ONLY
language. J




I will assume that rectangle ACDF is given as a golden rectangle, and I will
assume that the length of a side of square ABEF is " a ". the shorter side of
rectangle BCDE is given as " 1 ".  

By definition of 'golden rectangle':

matrix%281%2C2%2CSHORT%2CSIDE%29%2Fmatrix%281%2C2%2CLONG%2CSIDE%29%22%22=%22%22

AF%2FAC%22%22=%22%22AC%2F%28AF%2BAC%29
Rectangle BCDE is similar to rectangle ACDF.

matrix%281%2C4%2CSHORT%2CSIDE%2COF%2CBCDE%29%2Fmatrix%281%2C4%2CSHORT%2CSIDE%2COF%2CACDF%29%22%22=%22%22matrix%281%2C4%2CLONG%2CSIDE%2COF%2CBCDE%29%2Fmatrix%281%2C4%2CLONG%2CSIDE%2COF%2CACDF%29
a. Show that
(a/1) = 1/(a - 1)

BC%2FAF%22%22=%22%22BE%2FAC
1%2Fa%22%22=%22%22a%2F%28a%2B1%29
CROSS-MULTIPLY
a%5E2%22%22=%22%22a%2B1
a%5E2-a%22%22=%22%221  <-- use this in part (b)
a%28a-1%29%22%22=%22%221
Divide both sides by (a-1)
%28a%28a-1%29%29%2F%28a-1%29%22%22=%22%221%2F%28a-1%29
%28a%28cross%28a-1%29%29%29%2F%28cross%28a-1%29%29%22%22=%22%221%2F%28a-1%29
a%2F1%22%22=%22%221%2F%28a-1%29
b. Find the exact value of a(which will give you the golden ratio) by completing the square.

a%5E2-a%22%22=%22%221

1. Get half of coefficient of a:   (-1)/2 = -1/2
2. Square (-1/2), get +1/4
3. Add to both sides

a%5E2-a%2B1%2F4%22%22=%22%221%2B1%2F4

Factor left side into the square of a binomial:

%28a-1%2F2%29%28a-1%2F2%29%22%22=%22%225%2F4

%28a-1%2F2%29%5E2%22%22=%22%225%2F4

Take square roots of both sides, using ± on right.

sqrt%28%28a-1%2F2%29%5E2%29%22%22=%22%22%22%22+%2B-+sqrt%285%2F4%29

a-1%2F2%22%22=%22%22%22%22+%2B-+sqrt%285%29%2F2

a%22%22=%22%221%2F2+%2B-+sqrt%285%29%2F2

a%22%22=%22%22%281+%2B-+sqrt%285%29%29%2F2

Since ' a ' is not negative, we discard the - sign:

a%22%22=%22%22%281+%2B+sqrt%285%29%29%2F2

That is the exact value of ' a ', the golden ratio.

Edwin