Mathematics
ABC is an equilateral triangle. If AD bisects angle A, prove that AD is perpendicular bisector of BC.
Triangles
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Answer
Given: ABC is an equilateral triangle, and AD bisects ∠A.
To Prove: AD is the perpendicular bisector of BC.
Proof: Since AD bisects ∠A, we have:
∠BAD = ∠DAC
In Δ ABD and Δ ADC,
∠BAD = ∠DAC (Given, AD bisects ∠A)
AD = AD (Common side)
AB = AC (Sides of an equilateral triangle)
By SAS congruency, we conclude:
Δ ABD ≅ Δ ADC
Since Δ ABD ≅ Δ ADC, their corresponding parts are equal:
BD = DC
Thus, AD bisects BC.
Since the two triangles are congruent, their corresponding angles are equal:
∠ADB = ∠ADC
From the straight-line property:
∠ADB + ∠ADC = 180°
⇒ ∠ADB + ∠ADB = 180°
⇒ 2∠ADB = 180°
⇒ ∠ADB = = 90°
Thus, AD is perpendicular to BC.
Since AD bisects BC and is perpendicular to BC, we conclude that:
AD is the perpendicular bisector of BC.
Hence, AD is the perpendicular bisector of BC.
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