Mathematics Question and Solution
Calculating yellow area.
Let x be each of the three congruent red lengths.
Let y be each of the two congruent blue lengths.
a = (x+3) units.
It implies;
√(129)² = (x+3)²+x²+2x(x+3)cosb
129 = 2x²+6x+9+(2x²+6x)cosb
cosb = (120-6x-2x²)/(2x²+6x) --- (1).
Again.
y² = x²+3²+2*3*xcosb
y² = x²+9+6xcosb --- (2).
Again.
y² = (x+3)²+x²-2*x(x+3)cosb
y² = 2x²+6x+9-(2x²+6x)cosb --- (3).
Equating (2) and (3).
x²+9+6xcosb = 2x²+6x+9-(2x²+6x)cosb
6xcosb = x²+6x-(2x²+6x)cosb
6xcosb+(2x²+6x)cosb = x²+6x
(2x²+12x)cosb = x²+6x
2(x²+6x)cosb = x²+6x
2cosb = 1
cosb = 1/2
b = acos(1/2)
b = 60°
It implies;
c+b = 180
c = 180-60
c = 120°
Recall.
At (1).
cosb = (120-6x-2x²)/(2x²+6x)
And b = 60°
It implies;
cos60 = (120-6x-2x²)/(2x²+6x)
½ = (120-6x-2x²)/(2x²+6x)
Cross Multiply.
2(120-6x-2x²) = 2x²+6x
120-6x-2x² = x²+3x
3x²+9x-120 = 0
x²+3x-40 = 0
x²+8x-5x-40 = 0
x(x+8)-5(x+8) = 0
(x+8)(x-5) = 0
It implies;
x ≠ -8
x = 5 units.
Again, x is each of the three congruent red lengths.
Therefore, area yellow is;
½*3*xsinc
Where x = 5 units, c = 120° please.
= ½*3*5sin120
= ½*15*½√(3)
= ¼*15√(3)
= ¼(15√(3)) square units.
= 3.75√(3) square units.
= 6.49519052838 square units.
