Mathematics Question and Solution
Calculating length MN.
let x be the radius of the circle.
a = ½(15)
a = 7.5 units.
b = 24-a
b = 24-7.5
b = 16.5 units.
c²+b² = 24²
c = √(24²-16.5²)
c = 17.4284250579 units.
d²+a² = x²
d = √(x²-7.5²) units.
e = x+d
e = (x+√(x²-7.5²)) units.
Notice.
c = e
Calculating x.
17.4284250579 = x+√(x²-7.5²)
√(x²-7.5²) = 17.4284250579-x
x²-56.25 = (17.4284250579-x)²
x²-56.25 = 303.75-34.8568501158x+x²
-56.25 = 303.75-34.8568501158x
34.8568501158x = 303.75+56.25
34.8568501158x = 360
x = 360/34.8568501158
x = 10.3279555899 units.
Again, x is the radius of the circle.
f = ½(24)
f = 12 units.
f is AN =
BN.
g = f-a
g = 12-7.5
g = 4.5 units.
h = 24+a
h = 24+7.5
h = 31.5 units.
tanj = 17.4284250579/4.5
j = atan(17.4284250579/4.5)
j = 75.522487814°
j is angle ANM.
k = 180-j
k = 104.477512186°
k is angle BNM.
tanl = 17.4284250579/31.5
l = atan(17.4284250579/31.5)
l = 28.9550243718°
l is angle MBN.
m = 180-k-l
m = 180-104.477512186-28.9550243718
m = 46.5674634422°
m is angle BMN.
Therefore, length MN is, let it be y.
(y/sin28.9550243718) = (12/sin46.5674634422)
y = 8 units.
Again, y is length MN.
