Mathematics Question and Solution
Let the side length of triangle ABC be 1 unit.
a = 120-78
a = 42°
b = 60-42
b = 18°
(1/sin42) = (c/sin18)
c = 0.4618186516 unit.
tan60 = d/0.5
d = 0.86602540378 unit.
Area Green is;
0.5*0.4618186516*0.86602540378
= 0.19997334211 square units.
e = ⅓ unit.
Where e is the side length of the red inscribed regular hexagon.
Area Red is;
½*6*(⅓)²*(1/2tan(30))
= ⅓*(1/2tan(30))
= 0.28867513459 square units.
f = 60+b
f = 78°
g = 180-2(78)
g = 24°
h = 1+c
h = 1.4618186516 units.
Where h is the side length of the regular hexagon.
sin30 = i/h
i = 0.7309093258
j = 2(0.7309093258)+1.4618186516-1
j = 2.9236373032-1
j = 1.9236373032 units.
Area Blue is;
0.5(1.9236373032)²*sin24
= 0.7525401661 square units.
It implies;
Area Green : Area Red : Area Blue in the form 1 : a : b, where: a and b are to 2 decimal places is;
0.19997334211 : 0.28867513459 : 0.7525401661
= 1 : 1.44356808535 : 3.76320242568
≈ 1 : 1.44 : 3.76
