Mathematics Question and Solution
Calculating Alpha.
Let it be x.
Let 1 unit be the side length of the inscribed square.
Let r be the radius of the ascribed quarter circle.
a² = 1²+1²
a =√(2) units.
a is OE, the diameter of the inscribed square.
b = (r-1) units.
Calculating r.
1 ~ r
(r-1) ~ 1
Cross Multiply.
r(r-1) = 1
r²-r-1 = 0
(r-½)² = 1+(-½)²
(r-½)² = 1+¼
(r-½)² = (5/4)
r = ½±½√(5)
It implies;
r ≠ ½-½√(5)
r = ½+½√(5)
r = ½(1+√(5)) units.
r = 1.61803398875 units.
Again r is golden ratio, the radius of the ascribed quarter circle.
Recall.
b = r-1 units.
And r = 1.61803398875 units.
b = 1-1.61803398875
b = 0.61803398875 units.
It implies, angle alpha (x) is;
tanx = 1/b
x = atan(1/0.61803398875)
x = 58.2825255885°
Again, x is angle alpha.
