Question

In ∆ABC, ∠C is an obtuse angle. AD ⊥ BC and AB² = AC² + 3 BC². Prove that BC = CD.

Answer 1

Given: ΔABC where ∠C is an obtuse angle, AD ⊥ BC and AB² = AC² + 3BC²  

To prove: BC = CD

Proof: 

In ΔABC,  ∠C is obtuse.

Therefore,

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