Mathematics Question and Solution
Let the red length be a.
Calculating a.
b² = 8²+8²
b = 8√(2) units.
c² = 2²+2²
c = 2√(2) units
d = 8√(2)-2√(2)-2-½(a)
d = ½(12√(2)-4-a) units.
e² = 2a²-2a²cos120
e² = 3a²
e = √(3)a units.
Equating d and e to get a.
½(12√(2)-4-a) = √(3)a
12√(2)-4-a = 2√(3)a
12√(2)-4 = 2√(3)a+a
(12√(2)-4) = (2√(3)+1)a
a = (12√(2)-4)/(2√(3)+1)
a = ((12√(2)-4)(2√(3)-1))/11
a = (24√(6)-12√(2)-8√(3)+4)/11 units.
a = 2.9055258743 units.
Calculating Area Purple.
f = 180-120-45
f = 15°
(2.9055258743)² = 2g²
g = 2.0545170486 units.
h = 8-2.0545170486
h = 5.9454829514 units.
tan15 = j/5.9454829514
j = 1.5930873554 units.
Area Purple is;
0.5*1.5930873554*5.9454829514
= 4.7358368559 square units.
Calculating Area Orange.
k = 360-60-75-90
k = 135°
l = ½(180-135)
l = 22.5°
tan22.5 = 2/m
m = 4.8284271247 units.
n = 8-2-4.8284271247
n = 1.1715728753 units.
o = 4.8284271247- 0.5(2.9055258743)
o = 3.3756641876 units.
p² = 3.3756641876²+1.1715728753²-2*1.1715728753*3.3756641876cos135
p = 4.2849358529 units.
(4.2849358529/sin135) = (1.1715728753/sinq)
q = 11.1474623907°
r = 60-11.1474623907
r = 48.8525376093°
(2.9055258743/sin105) = (s/sin15)
s = 0.7785333116 units.
t = 0.7785333116+2.9055258743
t = 3.6840591859 units.
Area Orange is;
0.5*3.3756641876*1.1715728753sin135+0.5*3.6840591859*4.2849358529sin48.8525376093
= 1.3982458886+5.9435595759
= 7.3418054645 square units.
It implies;
Area Orange ÷ Area Purple to 2 decimal places is;
7.3418054645÷4.7358368559
= 1.5502657055
≈ 1.55
Therefore;
Exactly Red Length is;
= (24√(6)-12√(2)-8√(3)+4)/11 units.
= 2.9055258743 units.
Area Orange ÷ Area Purple to 2 decimal places is;
= 1.5502657055
≈ 1.55
