Circle AA, with a radius of 9, has a horizontal chord BC with a length of 10. From point C, a vertical line extends to point D. From point D, a horizontal line extends to point E on the circle's circumference. Line segment \DE has a length of 3. From point E, another vertical line extends to point F on the circle's circumference.
Points B andDD connect to form a line segment, and so do points D and F. What is the angle (in degrees) of ∠BDF?
Circle A, with a radius of 9, has a horizontal chord BC with a length of 10.
From point C, a vertical line extends to point D.
From point D, a horizontal line extends to point E on the circle's circumference.
Line segment DE has a length of 3.
From point E, another vertical line extends to point F on the circle's circumference.
Points B and D connect to form a line segment, and so do points D and F.
What is the angle (in degrees) of ∠BDF?
In △AEN:y24+82=92y24=92−+82y24=17y2=68y=2√17In △DEF:v2=32+y2v2=9+68v2=77v=√77In △BMA:52+z2=92z2=81−25z2=56z=2√14
x+y2=zx=z−y2x=2√14−2√172x=2√14−√17In △BCD:u2=102+x2u2=100+(2√14−√17)2u2=100+4∗14−4√14∗17+17u2=173−4√238u=√173−4√238
In △BPF:w2=132+(x+y)2w2=132+(2√14−√17+2√17)2w2=132+(2√14+√17)2w2=132+4∗14+4√14∗17+17w2=242+4√238
cos-rule:
In △BDF:w2=u2+v2−2uvcos(θ)cos(θ)=u2+v2−w22uvcos(θ)=173−4√238+77−(242+4√238)2√173−4√238√77cos(θ)=8−8√2382√13321−308√238cos(θ)=4(1−√238)√13321−308√238cos(θ)=−57.708994482292.5710938948cos(θ)=−0.62340188556θ=arccos(−0.62340188556)θ=128.564985997∘
The angle (in degrees) of ∠BDF is 128.564985997∘