Mathematics Question and Solution
Calculating Angle x.
Notice.
Quadrilateral ABCD is a square.
Let AB equal 2 units.
a = (2-y) units.
b = (1+y) units.
b is MQ.
c = (1-y) units.
Where y is the radius of the inscribed circle.
Calculating y.
(1+y)² = (1-y)²+(2-y)²
1+2y+y² = 1-2y+y²+4-4y+y²
4y = 4-4y+y²
y²-8y+4 = 0
(y-4)² = -4+(-4)²
(y-4)² = 12
y = 4±√(12)
Therefore;
y ≠ = (4+2√(3)) units.
y = (4-2√(3)) units.
y = 0.53589838486 units.
Again, y is the radius of the inscribed circle.
Recall.
b = (1+y)
And y = (4-2√(3)) units.
b = 1+4-2√(3)
b = (5-2√(3)) units.
a = (2-y)
And y = (4-2√(3)) units.
a = 2-(4-2√(3))
a = (2√(3)-2) units.
cosd = (2√(3)-2)/(5-2√(3))
d = 17.587953774°
d is angle BPM.
tan17.587953774 = 1/e
e = 3.15470053838 units.
e is BP.
f = e-2
f = 3.15470053838-2
f = 1.15470053838 units.
f is CP
Therefore, the required angle x is;
tanx = f/2
tanx = 1.15470053838/2
x = atan(1.15470053838/2)
x = atan(1/√(3))
x = 30°
