The arcs FA, AB, and <u>BF</u> make up the entire circle, so their measures sum to 360°:
arcFA + arcAB + arcBF = 360° → arcBF = 360° - 160° - 140° = 60°
By the inscribed angle theorem, angle BAF has measure equal to half of that of arc BF, so
angleBAF = 30°
By the secant-tangent theorem (not sure if there's an official name for it), the measure of angle ACF is equal to half the difference of the arcs it intercepts, namely BF and FA, so that
angleACF = 1/2 (arcFA - arcBF) = 1/2 (160° - 60°) = 50°
Now use the law of sines for solve for FA:
sin(angleACF) / FA = sin(angleBAF) / FC
sin(50°) / FA = sin(30°) / 10
FA = 10 sin(50°) / sin(30°) ≈ 15.3
