ABC is a triangle and D is the mid-point of BC. The perpendicular from D to AB and AC are equal. Prove that the triangle is isosceles.
Answers
Given: ABC is a triangle and D is the mid-point of BC i.e BD = DC . The perpendicular from D to AB and AC are equal i.e PD = DQ.
To prove: ABC is isosceles ∆.
PROOF :
In △BDP and △CDQ
PD = QD (Given)
BD = DC (Given)
∠BPD = ∠CQD (90°)
Therefore, △BDP ≅ △CDQ (by RHS congruence rule)
BP = CQ … ………..(1) (By CPCT)
In △APD and △AQD
PD = QD (Given)
AD = AD (common)
APD = AQD ( 90°)
Therefore, △APD ≅ △AQD (by RHS congruence rule)
So, PA = QA ……………... (2) (By CPCT)
On Adding eq (i) and (ii) :
BP + PA = CQ + QA
AB = AC
ABC is an isosceles ∆ since,Two sides of the triangle are equal.
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