Mathematics Question and Solution
Calculating Alpha.
Let it be a.
Let b be the radius of the ascribed quarter circle.
c = (b-√(2)) units.
d = (b-2√(2)) units.
It implies, observing similar plane shape (right-angled) side length ratios.
√(2) - (b-2√(2))
(b-√(2)) - b
Cross Multiply.
√(2)b = b²-2√(2)b-√(2)b+4
b²-4√(2)b+4 = 0
It implies;
b ≠ (2√(2)-2) units.
b = (2+2√(2)) units.
b = 4.8284271247 units
Again, b is the radius of the ascribed quarter circle.
Recall.
d = (b-2√(2)) units.
And b = (2+2√(2)) units.
d = (2+2√(2))-2√(2)
d = 2 units.
It implies;
sina = d/b
a = asin(2/(2+2√(2)))
a = asin(1/(1+√(2)))
a = 24.4698005207°
Again, a is the required angle alpha.
e = 90-a
e = 65.5301994793°
Area green is;
0.5*2*(2+2√(2))sin65.5301994793
= 4.3947364539 square units.
