Mathematics Question and Solution
Let r be the radius of the half circle.
a²+2² = r²
a = √(r²-4) units.
b = 2a
b = 2√(r²-4) units.
c = (r-2) units.
Calculating r.
√(r²-4) ~ 2√(r²-4)
(r-2) ~ 9
Cross Multiply.
9√(r²-4) = (2r-4)√(r²-4)
9 = 2r-4
2r = 13
r = ½(13) units.
r = 6.5 units.
Recall.
b = 2√(r²-4) units.
And r = ½(13) units.
b = 2√((½(13))²-4)
b = 2√(¼(169-16))
b = 2√(¼(153))
b = √(153) units.
b = 3√(17) units.
tand = 9/(3√(17))
d = atan(3/√(17))°
d = 36.0398934303°
e = (90-atan(3/√(17)))°
e = 53.9601065697°
f = ½(e)
f = 26.9800532848°
g = 90-½(d)
g = 71.9800532849°
It implies;
h² = (9+4)²+(6.5)²-2*13*6.5cos(71.9800532849)
h = 12.6083375488 units.
Let j be the radius of the green inscribed circle.
sinf = j/k
sin26.9800532848 = j/k
k = 2.20419542586j units.
Calculating j.
k+j+6.5 = h
2.20419542586j+j+6.5 = 12.6083375488
3.20419542586j = 6.1083375488
j = 1.90635611658 units.
Again, j is the radius of the green inscribed circle.
Therefore, Area Green Inscribed Circle is;
πj²
= π(1.90635611658)²
= 11.4171560513 square units.
