Mathematics Question and Solution
Calculating GH/HI.
Let 2 units be the side length of the regular pentagon.
Therefore, 1 unit is the radius of the circle.
a = ⅕*180(5-2)
a = 108°
a is the single interior angle of the regular pentagon.
b = 360-90-90-108
b = 180-108
b = 72°
b is angle DOH.
sinb = (HI)/(OD)
OD = 1 unit, radius of the circle.
sin72 = HI/1
HI = sin72 units.
HI = 0.9510565163 units.
Calculating GH.
c = 180-72
c = 108°
c is angle EOH.
d = 360-90-108-108
d = 270-216
d = 54°
d is angle EGH.
e = ½(180-108)
e = 36°
e is angle OEH equal angle OHE.
f = 90-e
f = 90-36
f = 54°
f is angle EHG.
Notice.
d = f = 54°
It implies;
g² = 1²+1²-2*1*1cos108
g = √(2-2cos108)
g = 1.61803398875 units.
g is EH equal EG.
h = 108-e
h = 108-36
h = 72°
h is angle GEH.
Therefore length GH is;
(GH)² = 1.61803398875²+1.61803398875²-2*1.61803398875*1.61803398875cos72
GH = 1.90211303259 units.
It implies;
GH/HI is;
1.90211303259/0.9510565163
= 2
