Question 9. In a certain circle, parallel chords of lengths $2, 3, 4$ correspond to central angles $\alpha, \beta, \alpha+\beta$. Here $\alpha+\beta<\pi$. If $\cos \alpha$ (which is a positive rational number) is expressed as a reduced (simplest) fraction, what is the sum of the numerator and the denominator?
Solution: Because in the same circle, equal chords subtend equal central angles, the parallelism of these chords is irrelevant. We can select points A, B, C such that AB = 2, BC = 3. Since \( A \widehat{C} = \alpha + \beta \), it follows that \( AC = 4 \). From \( \angle AC3 = \frac{\alpha}{2} \), using the cosine rule, we get

Thus, \( \cos \alpha = 2 \cos^2 \frac{\alpha}{2} - 1 = 2 \cdot \frac{49}{54} - 1 = \frac{17}{32} \). This is an irreducible fraction, so the sum of the numerator and the denominator is \( 17 + 32 = 49 \).