Question

In a triangle ABC, D is the mid-point of side AC such that BD = `1/2` AC. Show that ∠ABC is a right angle.

Answer 1

Given: In ΔABC, D is the mid-point of AC i.e., AD = CD such that BD = `1/2` AC.

To show: ∠ABC = 90°

Proof: We have BD = `1/2` AC  ...(i)

Since, D is the mid-point of AC.

∴ AD = CD = `1/2` AC  ...(ii)

From equations (i) and (ii),

AD = CD = BD

In ΔDAB, AD = BD   ...[Proved above]

∴ ∠ABD = ∠BAD   ...(iii) [Angles opposite to equal sides are equal]

In ΔDBC, BD = CD   ...[Proved above] 

∴ ∠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°  ...[From equations (iii) and (iv)]

⇒ ∠ABC + ∠ABC = 180°

⇒ 2∠ABC = 180°

⇒ ∠ABC = 90°