Question

In a triangle PQR, L and M are two points on the base QR, such that ∠LPQ = ∠QRP and ∠RPM = ∠RQP. Prove that:

1. ΔPQL ∼ ΔRPM
2. QL × RM = PL × PM
3. PQ² = QR × QL

Answer 1

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