Mathematics Question and Solution
Let DE = FB = a.
Let EG = GH = HF = b.
(3b)² = 12²+(12-2a)²
b² = ⅑(288-48a+4a²) ---- (1).
c = √(144+a²)
Where c is CF.
d = √((12-a)²-b²)
Where d is GC.
√(144+a²)² = (2b)²+√((12-a)²-b²)²
144+a² = 4b²+144-24a+a²-b²
3b² = 24a
b² = 8a ---- (2)
Equating (1) and (2).
8a = ⅑(288-48a+4a²)
72a = 288-48a+4a²
4a²-120a+288 = 0
a²-30a+72 = 0
(a-15)² = 225-72
a = 15±√(153)
a ≠ 15+3√(17)
a = 15-3√(17) cm.
a = 2.63068312315 cm.
Calculating b.
b² = 8a, and a = 2.63068312315 cm.
b² = 8*2.63068312315
b = 4.58753364949 cm.
AF = 12-2.63068312315
AF = 9.36931687685 cm.
e = 12-2*2.63068312315
e = 6.7386337537 cm.
tanf = 12/6.7386337537
f = 60.68348888734°
Where f is angle AFH.
(AH)² = 9.36931687685²+4.58753364949²-2*4.58753364949*9.36931687685cos60.68348888734
AH = 8.16937168659 cm.
(8.16937168659/sin60.68348888734) = (4.58753364949/sing)
g = 29.31651111271°
Where g is angle FAH.
Angle DAH is;
90-29.31651111271
= 60.68348888729°
Therefore;
Purple Zone Area is;
0.5*12*8.16937168659sin60.68348888729
= 42.73863375368 cm²
