Question

PQRS is a quadrilateral in which PQ parallel SR and PR, QS intersect each other On. prove that PO/RO=QO/SO

Answer 1

Answer:

Proved

Step-by-step explanation:

PQRS is a quadrilateral in which PQ parallel SR and PR, QS intersect each other On. prove that PO/RO=QO/SO

Let see two triangles

ΔPOQ    &  ΔROS

As PQ ║ SR so

∠OQP  =  ∠OSR

∠OPQ = ∠ORS

in Δ POQ

PO/Sin(∠OQP)  = QO/Sin(∠OPQ)  

=> Sin(∠OPQ) / Sin(∠OQP) = PO/QO   - Eq 1

Sumilalry in ΔROS

RO/Sin(∠OSR)  = SO/Sin(∠ORS)

Raplacing  ∠OSR with ∠OQP   & ∠ORS with ∠OPQ

RO/Sin(∠OQP)  = SO/Sin(∠OPQ)

=>  Sin(∠OPQ)/Sin(∠OQP)  = RO/SO  - Eq 2

Equating Eq 1 & Eq 2

PO/QO = RO/SO

=> PO/RO = QO/SO

QED

Answer 2

Answer:

Step-by-step explanation:

Consider, two triangles

1) ΔPOQ    

2) ΔROS

Since PQ ║ SR ,

∠OQP  =  ∠OSR

and

∠OPQ = ∠ORS

For Δ POQ

PO/Sin(∠OQP)  = QO/Sin(∠OPQ)  

=> Sin(∠OPQ) / Sin(∠OQP) = PO/QO   - ---- 1

For ΔROS

RO/Sin(∠OSR)  = SO/Sin(∠ORS)

Replacing  ∠OSR with ∠OQP   & ∠ORS with ∠OPQ

RO/Sin(∠OQP)  = SO/Sin(∠OPQ)

=>  Sin(∠OPQ)/Sin(∠OQP)  = RO/SO   ----- 2

Equating 1 & 2

PO/QO = RO/SO

=> PO/RO = QO/SO