Mathematics Question and Solution
Let radius of the half circle be 2 units.
sin22.5 = d/2
d = 0.76536686473 units.
BZ = 2d
BZ = 1.53073372946 units.
e = 90-½(180-45)
e = 22.5°
cos22.5 = f/4
f = 3.69551813005 units.
f is CZ.
cos67.5 = g/1.53073372946
g = 0.58578643763 units.
h = 2g
h = 1.17157287525 units.
j = 0.5(2-1.17157287525)
j = 0.41421356238 units.
CX = 2+0.41421356238
CX = 2.41421356238 units.
Calculating c.
tan22.5 = c/2.41421356238
c = 1 unit.
Calculating b.
2² = k²+0.41421356238²
k = 1.95663668696 units.
k = b+c
Therefore;
b = 1.95663668696-1
b = 0.95663668696 units.
CH = √(1+2.41421356238²)
CH = 2.61312592976 units.
HZ = CZ-CH
HZ = 3.69551813005-2.61312592976
HZ = 1.08239220029 units.
Calculating a.
sin22.5 = (HZ)/l
sin22.5 = 1.08239220029/l
l = 2.82842712474 units.
Where l = b+a (AH).
Therefore;
a = l-b
a = 2.82842712474-0.95663668696
a = 1.87179043778 units.
Notice;
a = 1.87179043778 units.
b = 0.95663668696 units
c = 1 unit.
Therefore, observing the prove.
(a/b)-(b/c) = 1
(1.87179043778/0.95663668696)-(0.95663668696/1)
= 1.95663668694-0.95663668694
= 1
Proved!
