Mathematics Question and Solution
x² = (r+1)² - (r-1)²
Therefore x = 2√(r) unit.
y² = (r+1)² - (2√(r))²
Therefore y = √(r²-2r+1)
y = √(r-1)²
y = (r-1) unit.
z = (r+1) - y
z = (r+1) - (r-1)
Therefore z = 2 units.
a = r+r
Therefore a = 2r units.
A sincere study and analysis of the composite circles deduced a right-angled triangle with two adjacent sides, 2 units and 2√(r) units respectively, and hypotenuse 2r units.
Therefore;
Calculating r, the radius of the three congruent circles.
(2r)² = 2² + (2√(r))²
4r² = 4 + 4r
4r² - 4r - 4 = 0
r² - r - 1 = 0
Resolving the above quadratic equation via completing the square side to get r, radius of the three congruent circles.
(r - ½)² = 1 + (-½)²
(r - ½)² = 1 + ¼
(r - ½)² = ¼(5)
r - ½ = ±√(5/4)
r = ½ ± ½√(5)
r = ½(1±√(5))
It implies;
r ≠ ½(1-√(5))
r = ½(1+√(5)) units.
r = 1.6180339887 units.
