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PQRS is a trapezium with $PQ \parallel SR$. U and V are points on the non-parallel sides PS and QR respectively as shown in the given figure. If $UV \parallel SR$, prove that $\frac{PU}{US} = \frac{QV}{VR}$.
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Solution:
Construction: Join PR intersecting UV at X (1/2 Mark)
In $\Delta PSR$, $UX \parallel SR$, $\frac{PU}{US} = \frac{PX}{XR}$----(1) (1 Mark)
In $\Delta PQR$, $XV \parallel PQ$, $\frac{PX}{XR} = \frac{QV}{VR}$----(2) (Given $PQ \parallel SR$ and $UV \parallel SR$) (1 Mark)
From (1) and (2), $\frac{PU}{US} = \frac{QV}{VR}$ (1/2 Mark)
Construction: Join PR intersecting UV at X (1/2 Mark)
In $\Delta PSR$, $UX \parallel SR$, $\frac{PU}{US} = \frac{PX}{XR}$----(1) (1 Mark)
In $\Delta PQR$, $XV \parallel PQ$, $\frac{PX}{XR} = \frac{QV}{VR}$----(2) (Given $PQ \parallel SR$ and $UV \parallel SR$) (1 Mark)
From (1) and (2), $\frac{PU}{US} = \frac{QV}{VR}$ (1/2 Mark)