Mathematics Question and Solution
Let a = 1 unit.
It implies;
1 ~ b
b ~ e
Cross Multiply.
e = b²
f = (1-b²) units.
cos60 = g/(1-b)
g = ½(1-b) units.
h = 2g+1
h = (2-b) units.
It implies;
Calculating b.
(2-b)² = (1-b²)²+1²-2(1-b²)cos120
4-4b+b² = 1-2b²+b⁴+1+1-b²
1-4b+4b² = b⁴
b⁴+0b³-4b²+4b-1 = 0
It implies;
b ≠ 1 unit.
b ≠ -(1+√(2)) units.
b = (√(2)-1) units.
b = 0.41421356237 units.
Recall.
e = b²
e = (√(2)-1)²
e = (3-2√(2)) units
e = 0.17157287525 units.
It implies;
Calculating c.
c² = (√(2)-1)²+1²-2(√(2)-1)cos120
c² = 3-2√(2)+1+√(2)-1
c = √(3-√(2)) units.
c = 1.25928012675 units.
Calculating d.
d² = 0.414213562372²+0.17157287525²-2*0.414213562372*0.17157287525cos120
d = 0.52161090732 units.
The Proof.
a÷b is;
1÷0.41421356237
= 2.41421356239
(c+d)÷(c-d) is;
(1.25928012675+0.52161090732)÷(1.25928012675-0.52161090732)
= 2.41421356234
It implies;
(a÷b) = ((c+d)÷(c-d))
2.41421356239 = 2.41421356234
Proved.
