Mathematics Question and Solution
Calculating x, length DF.
Let a be the radius of the ascribed half circle.
b = (a-6) units.
c = (a-7) units.
7² = 6²+8²-2*6*8cosd
96cosd = 36+64-49
d = acos((36+64-49)/96)
d = 57.9100487437°
d is angle ACB.
e = 180-d
e = 122.089951256°
e is angle ACE.
Calculating a.
a² = 8²+(a-6)²-2*8(a-6)cos122.089951256
a² = 64+a²-12a+36+8.5a-51
3.5a = 49
a = 14 units.
Again, a is the radius of the half circle.
Recall.
c = (a-7) units.
And a = 14 units.
c = 14-7
c = 7 units.
It implies;
9² = 7²+14²-2*7*14cosf
196cosf = 49+196-81
f = acos((49+196-81)/196)
f = 33.203099198°
Therefore, the required length x, (length DF) is;
x² = 2(14)²-2(14)²cos33.203099198
x² = 63.9999999991
x = √(63.9999999991)
x = 8 units.
