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Question
In the following figure, write BC, AC, and CD in ascending order of their lengths.
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Solution
In ΔABC,
AB = AC
⇒ ∠ABC = ∠ACB ..(angles opposite to equal sides are equal)
⇒ ∠ABC = ∠ACB = 67°
⇒ ∠BAC = 180° - ∠ABC - ∠ACB ...(Angle sum property)
⇒ ∠BAC = 180° - 67° - 67°
⇒ ∠BAC = 46°
Since ∠BAC < ∠ABC, we have
BC < AC ...(1)
Now, ∠ACD = 180° - ACB ...(Linear pair)
⇒ ∠ACD = 180° - 67°
⇒ ∠ACD = 113°
Thus, in ΔACD,
∠CAD = 180°- ∠ACD + ∠ADC
⇒ ∠CAD = 180° - (113° + 33°)
⇒ ∠CAD = 180° - 146°
⇒ ∠CAD = 34°
Since ∠ADC < ∠CAD, we have
AC < CD ...(2)
From (1) and (2), we have
BC < AC < CD
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