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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