Mathematics Question and Solution
Let the three equal lengths be a.
Radius r of the semi circle is;
r = 5+4
r = 9 units.
5² = b²+a²-2abcos45
25 = b²+a²-√(2)ab --- (1).
9² = b²+a²-2abcos135
81 = b²+a²+√(2)ab --- (2).
Solving (1) and (2) simultaneously. Subtracting (1) from (2).
81 = b²+a²+√(2)ab
-(25 = b²+a²-√(2)ab)
56 = 2√(2)ab
28 = √(2)ab
a = 28/√(2)b
a = 14√(2)/b --- (3).
Substituting (3) in (1) to get b.
25 = b²+(14√(2)/b)²-√(2)(14√(2)/b)b
25 = b²+(392/b²)-28
53 = b²+(392/b²)
53 = (b⁴+392)/b²
53b² = b⁴+392
b⁴-53b²+392 = 0
Let b² = p
It implies;
p²-53p+392 = 0
Resolving the above quadratic equation via completing the square approach.
p²-53p+392 = 0
(p-½(53))² = -392+(-½(53))²
(p-½(53))² = -392+¼(2809)
(p-½(53))² = ¼(2809-1568)
(p-½(53))² = ¼(1241)
p-½(53) = ±√(¼(1241))
p-½(53) = ±½√(1241)
p = ½√(53)±½√(1241)
p = 3.6400549446±17.6139149538
It implies;
p ≠ 3.6400549446-17.6139149538
p = 3.6400549446+17.6139149538
p = 21.2539698984 units.
Notice!
p = b²
b² = 21.2539698984
b = √(21.2539698984)
b = 4.6102028045 units.
Calculating a, using (1).
a = 14√(2)/b
And b = 4.6102028045 units.
a = 14√(2)/4.6102028045
a = 4.2946028001 units.
(5/sin45) = (4.6102028045/sinc)
c = 40.6911378916°
d = 2a
d = 2*4.2946028001
d = 8.5892056002 units.
Therefore, x is;
sin40.6911378916° = x/8.5892056002
x = 8.5892056002sin40.6911378916
x = 5.6 units.
