CBSE Class 9 Answered

Given: D is the midpoint of BC and PD = DQ

To prove: ABC is isosceles

Proof:

In _{BDP and CDQ}

_{BD = DC (D is mid-point)}

_{PD = QD (given)}

_{BPD = CQD (each 90 degrees)}

_{Therefore, BDP CDQ (By RHS)}

_{So, BP = CQ … (1) (By CPCT)}

Again in _{APD and AQD}

_{AD = AD (common)}

_{PD = QD (given)}

_{APD = AQD (each 90 degrees)}

_{Therefore, APD AQD (By RHS)}

_{So, PA = QA … (2) (By CPCT)}

_{Adding (1) and (2), we get}

_{BP + PA = CQ + QA}

_{Or, AB = AC.}