Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Notice;
Side length of the congruent regular hexagon is 2 units.
(a) Calculating
Length Blue ÷ Length Red.
Length Blue is;
Let it be x.
4²=2²+x²-4xcos150
x²+2√(3)x-12= 0
It implies;
x (Length Blue) = (√(15)-√(3)) units.
Length Red is;
Let it be y.
y = √(8-8cos120)-x
y = 2√(3)-(√(15)-√(3))
It implies;
y (Length Red) = (3√(3)-√(15)) units.
Therefore;
Length Blue ÷ Length Red exactly is;
(√(15)-√(3)) ÷ (3√(3)-√(15))
= ½(√(5)-1)
(b) Calculating Shaded Area exactly in decimal.
It is;
Area regular hexagon with side 2 units + Area equilateral triangle with side 2 units - Area triangle with height 2 units and base 2sin120 units - Area triangle with height (√(15)-√(3)) units and base 2sin150 units - Area sector with radius 4 units and angle 44.47751218593°.
= 6√(3) + √(3) - √(3) - ½(√(15)-√(3)) - 6.21023224592
= 6√(3) - 1.07046626932 - 6.21023224592
= 3.11160633017 square units.
