Question : Chords $\overline{AB}$ and $\overline{CD}$ of a circle intersect inside the circle at point F. If $m(\overline{AF})=2.4$ cm, $m(\overline{BF})=1.8$ cm and $m(\overline{CD}) = 5.7$ cm, what is the length (in cm) of the longer of the two line segments, $\overline{CF}$ and $\overline{DF}$?
Option 1: 4.5
Option 2: 4.8
Option 3: 5.4
Option 4: 3.6
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Correct Answer: 4.8
Solution : $m(\overline{AF}) = 2.4$ cm $m(\overline{BF}) = 1.8$ cm $m(\overline{CD}) = 5.7$ cm Let CF = $x$ cm DF = ($5.7-x$) cm Since AB and CD are two chords of a circle that intersect at point F, AF × BF = CF × DF ⇒ 2.4 × 1.8 = $x$ × ($5.7-x$) $\therefore x$ = 4.8, 0.9 CF = 4.8, DF = 0.9 or, CF = 0.9, DF = 4.8 $\therefore$ Longest value = 4.8 Hence, the correct answer is 4.8.
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