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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.
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Solution
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°
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