jmc

Question

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of , , and + radians, respectively, where + < . If , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Accepted Answer

FIGURE It’s easy to see in triangle which lengths 2, 3, and 4, that the angle opposite the side 2 is 2, and using the Law of Cosines, we get:2^2 = 3^2 + 4^2 - 2342Which, rearranges to:21 = 242And, that gets us:2 = 7/8Using 2 = 2^2 - 1, we get that: = 17/32Which gives an answer of 49.