Question : D and E are two points on the sides AC and BC, respectively of $\triangle ABC$ such that DE = 18 cm, CE = 5 cm, and $\angle$DEC = 90º. If $ \tan\angle$ABC = 3.6, then AC : CD = ?
Option 1: BC : 2CE
Option 2: 2CE : BC
Option 3: 2BC : CE
Option 4: CE : 2BC
Correct Answer: 2BC : CE
Solution :
Based on the question, we draw a figure of $\triangle$ ABC,
$\angle$DEC = 90°
DE = 18 cm
CE = 5 cm
⇒ $\tan\angle$DCE $=\frac{DE}{CE}=\frac{18}{5}=$ 3.6
⇒ $\tan\angle$ABC = 3.6
⇒ $\angle$DCE = $\angle$ABC
$\therefore$ AC = AB
$\angle$DCE + $\angle$CDE = 90°
⇒ 2$\angle$DCE + 2$\angle$CDE = 180°
Also, $\angle$DCE + $\angle$CAB + $\angle$ABC = 180°
⇒ 2$\angle$DCE + $\angle$CAB = 180°
$\therefore$ $\angle$CAB = 2$\angle$CDE
⇒ $\frac{AC}{CB} = \frac{2CD}{CE}$
⇒ $\frac{AC}{CD} = \frac{2CB}{CE} =$ 2BC : CE
Hence, the correct answer is 2BC : CE.
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