Mathematics Question and Solution
Calculating angle HCP.
Let it be x.
Let 1 unit be the radius of the inscribed circle.
It implies;
2 units is the diameter of the diameter of the inscribed circle and also the side length of the ascribed square.
a² = 2(1²)
a² = 2
a = √(2) units.
a is half the diagonal of the ascribed square.
b = 1+a
b = (1+√(2)) units.
b = 2.41421356237 units.
c² = 2.41421356237²+1²-2*2.41421356237*1cos45
c = 1.84775906502 units.
(1.84775906502/sin45) = (1/sind)
d = 22.5°
e = 180-d-45
e = 180-22.5-45
e = 112.5°
e is angle DQH.
f = 45+d
f = 67.5°
f is angle CQH.
g = f-45
g = 67.5-45
g = 22.5°
cos22.5 = h/√(2)
h = √(2)cos22.5
h = 1.30656296488 units.
h is HQ.
j² = 1.30656296488²+1²-2*1.30656296488*1cos67.5
j = 1.30656296488 units.
j is CH.
It implies;
h = j
Meaning angle CQH = angle HCQ = 67.5°
Therefore, the required angle x is;
90-f
x = 90-67.5
x = 22.5°
