Question

PQRS is a trapezium with PQ || SR. U and V are points on the non-parallel sides PS and QR respectively as shown in the given figure. If UV || SR. prove that `(PU)/(US) = (QV)/(VR)`.

Answer 1

Given:

Trapezium PQRS with PQ || SR.

UV || SR

Since PQ || SR and UV || SR, we can conclude that all three lines are parallel: PQ || UV || SR.

To Prove: `(PU)/(US) = (QV)/(VR)`

Construction:

Join the diagonal PR. Let it intersect UV at point O.

Proof:

1. In ΔPSR:

We know that UO || SR (since UV || SR).

By the Basic Proportionality Theorem (BPT):

`(PU)/(US) = (PO)/(OR)`   ...(Equation 1)

2. In ΔPRQ:

We know that OV || PQ (since UV || PQ).

By the Basic Proportionality Theorem (BPT):

`(QV)/(VR) = (PO)/(OR)`   ...(Equation 2)

3. From equation 1 and equation 2, both ratios are equal to `(PO)/(OR)`.

Therefore, `(PU)/(US) = (QV)/(VR)`

Hence Proved.