Mathematics Question and Solution
Let OA be 1 unit.
Area Triangle AOB is;
½*1*1
= ½ square units.
Calculating Area Triangle ODG.
a² =2(1)²-2(1)²cos30
a = √(2-√(3)) units.
a = 0.51763809021 units.
a is AC, the side length of the big square.
b² = 2(0.51763809021)²
b = 0.73205080758 units.
b is AD, the diagonal of the big square.
c² = 1²+0.73205080758²-2*0.73205080758cos120°
c = 1.50597117916 units.
c is OD.
(1.50597117916/sin120) = (1/sind)
d = 35.1039093611°
d is angle ADO
e = 180-35.1039093611-120
e = 24.8960906389°
e is Ange AOD.
Notice.
BC = 1 unit.
BC is the side length of the bigger square.
f² = 2(1)²
f = √(2) units.
f = 1.41421356237 units.
f is BG.
g² = 2+1-2√(2)cos105
g = 1.93185165258 units.
g is OG.
(1.93185165258/sin105) = (√(2)/sinh)
h = 45°
h is angle BOG.
j = 90+e-h
j = 90-24.8960906389-45
j = 20.1039093611°
j is angle DOG.
Therefore;
Area triangle ODG is;
0.5*1.50597117916*1.93185165258sin20.1039093611
= ½ square units.
It implies;
Area triangle AOB, ½ square units is equal Area triangle ODG, ½ square units as well.
Proved.
