Mathematics Question and Solution
Let AC be 2 units.
It implies;
OA = AB = BC = DE = EF = 1 unit.
Blue length = 1 unit.
Calculating red length.
a² = 5-4cos72
a = 1.94008557092 units.
a is OD.
(1.94008557092/sin72) = (1/sinb)
b = 29.35462809223°
b is angle ADO.
c = 108+29.35462809223
c = 137.35462809223°
c is angle ODE.
Calculating OE.
(OE)² = 1.94008557092²+1-2*1.94008557092cos137.35462809223
OE = 2.76007862003 units.
OE is the radius of the inscribed arc.
(2.76007862003/sin137.35462809223) = (1/sind)
d = 14.20837250736°
d is angle DOE.
e = 180-72-29.35462809223
e = 78.64537190777°
e is angle AOD.
f = 78.64537190777-14.20837250736
f = 64.43699940041°
f is angle EOC, the angle of the inscribed arc.
sing = (2.76007862003/3)
g = 66.92991347173°
h = 108-66.92991347173
h = 41.07008652827°
j = 360-41.07008652827-108-108
j = 102.92991347173°
k = 180-102.92991347173-72
k = 5.07008652827°
l² = 8-8cos108
l = 3.2360679775 units.
l is CF.
Therefore, Red Length is;
Let it be m.
(m/sin5.07008652827) = (3.2360679775/sin102.92991347173)
m = 0.29342506056 units.
Therefore;
Length Red ÷ Length Blue to 2 decimal places is;
0.29342506056÷1
= 0.29342506056
≈ 0.29
