Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let a be the equal lengths inscribing triangle ABC.
Calculating a.
½(a)² = 15
a = √(30) units.
b = 180-60-45
b = 75°
b is angle BAC.
c = 180-2b
c = 30°
d² = 2√(30)²
d = √(60)
d = 2√(15) units.
e = 180-60-45
e = 75°
√(30)/sin60 = (f/sin45)
2√(10) = √(2)f
f = 2√(5) units.
g = d+f
g = 2(√(15)+√(5)) units.
Area ABC is;
½*2√(30)*2(√(15)+√(5))sin45
= √(30)*(√(15)+√(5))*√(2)
= 2√(15)(√(15)+√(5))
= 30+10√(3) square units.
Yellow Area is;
Area ABC - Area triangle with height and base √(30) units respectively - Area isosceles triangle with height √(30) units and base √(30)sin30 units.
= (30+10√(3))-½(√(30)*√(30)-½(√(30)*√(30)sin30)
= (30+10√(3))-15-½(15)
= 15-½(15)+10√(3)
= ½(15+20√(3)) square units.
Therefore;
Yellow Area ÷ Area ABC exactly is;
½(15+20√(3))÷(30+10√(3)
= (3+4√(3))/(12+4√(3))
= ⅛(3√(3)-1) exactly.
= 0.52451905284 in decimal.
