Mathematics Question and Solution
Calculating yellow inscribed circle area.
a² = 6²+8²
a = √(100)
a = 10 units.
2b² = 10²
b² = 50
b = 5√(2) units.
b is the radius of the half circle.
(5√(2)/sin45) = (8/sinc)
c = 53.1301023542°
d = 180-c
d = 126.8698976458°
e = 180-45-d
e = 8.1301023542°
(5√(2)/sin45) = (f/sin8.1301023542)
f = 1.4142135624 units.
f = √(2) units.
Let g be the radius of the inscribed yellow circle.
h = (5√(2)-g) units.
j² = 2g²
j = √(2)g units.
Calculating g, radius of the inscribed yellow circle.
(5√(2)-g)² = √(2)²+(√(2)g)²
50-10√(2)g+g² = 2+2g²
g²+10√(2)g-48 = 0
Resolving the above quadratic equation via completing the square approach to get g, radius of the inscribed yellow circle.
g²+10√(2)g-48 = 0
(g+5√(2))² = 48+(5√(2))²
(g+5√(2))² = 98
g = -5√(2)±√(98)
g = -5√(2)±7√(2)
It implies;
g ≠ -7√(2)-5√(2)
g = 7√(2)-5√(2)
g = 2√(2) units.
Again, g is the radius of the inscribed yellow circle.
Therefore, area yellow inscribed circle is;
πg²
= π(2√(2))²
= 8π square units.
