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प्रश्न
In a triangle PQR, L and M are two points on the base QR, such that ∠LPQ = ∠QRP and ∠RPM = ∠RQP. Prove that:
- ΔPQL ∼ ΔRPM
- QL × RM = PL × PM
- PQ2 = QR × QL
उत्तर
i. In ΔPQL and ΔRMP
∠LPQ = ∠QRP ...(Given)
∠RQP = ∠RPM ...(Given)
ΔPQL ∼ ΔRMP ...(AA similarity)
ii. As ΔPQL ∼ ΔRMP ...(Proved above)
`(PQ)/(RP) = (QL)/(PM) = (PL)/(RM)`
`=>` QL × RM = PL × PM
iii. ∠LPQ = ∠QRP ...(Given)
∠Q = ∠Q ...(Common)
∆PQL ∼ ∆RQP ...(AA similarity)
= `(PQ)/(RQ) = (QL)/(QP) = (PL)/(PR) `
`=>` PQ2 = QR × QL
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