Mathematics
In △ABC, AB = AC and D is a point in side BC such that AD bisects angle BAC.
Show that AD is perpendicular bisector of side BC.
Triangles
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Answer
Given: ABC is a triangle such that D is a point on side BC and ∠BAD = ∠ CDA.
To prove: AD ⊥ bisector of BC.
Proof: In Δ ABD and Δ ACD,
∠BAD = ∠CAD (Given)
AD = AD (Common Side)
AB = AC (Given)
Thus, by SAS congruency criteria,
Δ ABD ≅ Δ ACD
By corresponding parts of congruent triangles,
BD = CD and ∠ADB = ∠ADC
As ∠ADB and ∠ADC form linear pair.
⇒ ∠ADB + ∠ADC = 180°
⇒ ∠ADB + ∠ADB = 180°
⇒ 2∠ADB = 180°
⇒ ∠ADB =
⇒ ∠ADB = 90° = ∠ADC
This proves AD is perpendicular to BC.
Since AD is perpendicular to BC and BD = DC, AD is the perpendicular bisector of BC.
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