Question 13 In a triangle ABC, D is the mid-point of side AC such that BD = 12 AC. Show that ∠ABC=90∘.
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Solution
We have, BD=12AC...(i) D is the mid-point of AC. ∴AD=CD=12AC...(ii) From Eqs. (i) and (ii), AD = CD = BD In ΔADB,AD=BD[provedabove] ∴∠ABD=∠BAD...(iii) [angles opposite to equal sides are equal] In ΔDBC,BD=CD[provedabove]∴∠BCD=∠CBD...(iv) [angles opposite to equal sides are equal] In ΔABC,∠ABC+∠BAC+∠ACB=180∘ [by angle sum property of a triangle] ⇒∠ABC+∠BAD+∠DCB=180∘⇒∠ABC+∠ABD+∠CBD=180∘fromEqs.(iii)and(iv)]⇒∠ABC+∠ABC=180∘⇒∠ABC=90∘