Mathematics Question and Solution
Calculating Area Green.
Let x be the side length of the ascribed square.
a²+3² = x²
a = √(x²-9) units.
b = (90-y)°
siny = 3/x --- (1).
7² = x²+√(x²-9)²-2x√(x²-9)cos(90-y)
49 = 2x²-9-2x√(x²-9)siny
58 = 2x²-2x√(x²-9)siny
29 = x²-x√(x²-9)siny --- (2).
Substituting (1) in (2).
29 = x²-x√(x²-9)(3/x)
29 = x²-3√(x²-9)
3√(x²-9) = x²-29
3²(x²-9) = (x²-29)
9x²-81 = x⁴-58x²+841
x⁴-67x²+922 = 0
(x²-½(67))² = -922+(-½(67))²
(x²-½(67))² = 801/4
x² = ½(67)±√(801/4)
x²= ½(67)±½(3√(89))
It implies;
x² = ½(67-3√(89))
Or
x² = ½(67+3√(89))
Therefore;
x ≠ √(½(67-3√(89))) units.
x = √(½(67+3√(89))) units.
x = 6.90348003452 units.
Again, x is the side length of the ascribed square.
Recall.
a = √(x²-9) units.
And x = 6.90348003452 units.
a = √(6.90348003452²-9)
a = 6.21755873209 units.
Calculating y.
tany = 3/a
y = atan(3/6.21755873209)
y = 25.7575178503°
z = 90-y
z = 90-25.7575178503
z = 64.2424821497°
It implies, area green is;
Area square with side length 6.90348003452 units - Area triangle with height 6.21755873209 units and base 3 units - Area triangle with height 6.21755873209 units and base 6.90348003452sin64.2424821497 units.
6.90348003452²-0.5(3*6.21755873209)-0.5(6.21755873209*6.90348003452sin64.2424821497)
= 47.658036587-9.32633809814-19.3290182935
= 19 square units.
