Mathematics Question and Solution
Let AC be 1 unit.
Area smaller pentagon is;
½(5)*(1/(2tan(36)))
= 1.72047740059 square units.
a² = 1²+0.5²-cos108
a = 1.24860602048 units.
Where a is CD.
(1.24860602048/sin108) = (0.5/sinb)
b = 22.38617755917°
c = 108-22.38617755917
c = 85.61382244083°
Calculating Area E.
It is;
0.5*0.5*0.5*1.24860602048sin22.38617755917
= 0.05944103227 square units.
(a) Area E ÷ Area Smaller Pentagon to 3 decimal places is;
0.05944103227 ÷ 1.72047740059
= 0.03454915028
≈ 0.035
Calculating r, radius of the inscribed circle.
(r/tan(0.5*85.61382244083)) + (r/tan(54)) = 1
1.80618303497r = 1
r = 0.55365374419 unit.
(b) Radius : AB : CD in the form, 1 : a : b, where: a and b are to 2 decimal places is;
0.55365374419 : 1 : 1.24860602048
= 1 : 1.80618303497 : 2.25521101155
≈ 1 : 1.81 : 2.26
Where;
a = 1.81
b = 2.26
