In ∆ABC, ∠C is an obtuse angle. AD ⊥ BC and AB2 = AC2 + 3 BC2. Prove that BC = CD.
Given: ΔABC where ∠C is an obtuse angle, AD ⊥ BC and AB2 = AC2 + 3BC2
To prove: BC = CD
In ΔABC, ∠C is obtuse.
AB2 = AC2 + BC2 + 2BC × DC (Obtuse angle theorem) …(1)
AB2 = AC2 + 3BC2 (Given) …(2)
From (1) and (2), we get
AC2 + 3BC2 = AC2 + BC2 + 2BC × DC
⇒ 3BC2 = BC2 + 2BC × DC
⇒ 2BC2 = 2BC × DC
⇒ BC = DC
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