Mathematics Question and Solution
a² = 3²+4²
a = √(25)
a = 5 cm.
a is the radius of the quarter circle.
Let r be the radius of the inscribed blue circle.
Calculating r.
b = (3+r) cm.
c = (5-r) cm.
It implies;
(5-r)² = (3+r)²+r²
25-10r+r² = 9+6r+r²+r²
r²+16r-16 = 0
Resolving the above quadratic equation via completing the square approach to get r, radius of the inscribed blue circle.
(r+8)² = 16+(8)²
(r+8)² = 80
r = -8±√(80)
r = -8±4√(5)
It implies;
r = (4√(5)-8) cm.
r = 0.94427191 cm.
Therefore, area blue inscribed circle is;
πr²
= π(4√(5)-8)²
= π(80-64√(5)+64)
= π(144-64√(5))
= 16π(9-4√(5)) cm²
= 2.8011993303 cm²
