Problem:
In a circle, parallel chords of lengths 2,3 and 4 determine central angles of α,β and α+β radians, respectively, where α+β<π. If cosα, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
Solution:
Since any two chords of equal length subtend equal angles, the parallelism of the chords is irrelevant. Therefore, we may choose points A,B and C, as shown in the first accompanying figure, so that AB=2,BC=3 and (since AC=α+β)AC=4. Since ∠ACB=2α, by the Law of Cosines we have
cos2α=2⋅AC⋅BCAC2+BC2−AB2=2⋅4⋅342+32−22=87
Therefore, one finds that
cosα=2cos22α−1=2⋅6449−1=3217
and that the desired answer is 17+32 or 49.
OR
Attacking the problem more directly, let r denote the radius of the circle, and note that sin2α=r1, as shown in the second figure above. One similarly finds that sin2β=2r3 and sin2α+β=r2. Since sin(2α+2β)=sin2αcos2β+cos2αsin2β, it follows that
r2=r11−4r29+2r31−r21(1)
Solving this for r21, one finds that it is equal to 6415. Upon checking that this root is not extraneous to (1), it follows that
