Question

PQRS is a parallelogram and O is a point on SQ. The produced line PO meets QR at T and SR produced at U. If SO=3OQ, then find the value of PQ/RU​

Answer 1

Answer:1/2

Step-by-step explanation:

Answer 2

Answer:

Option B. $\frac{1}{2}$ is the correct option.

Step-by-step explanation:

In a parallelogram PQRS,

SP ║ RQ and SQ is a transverse,

Therefore, ∠SQR ≅ ∠PSQ  [Alternate angles]

SP ║ RQ and PT is a transverse,

Therefore, ∠SOP ≅ ∠TOQ [Vertically opposite angles]

By (AA) property both the triangles ΔSOP and ΔTOQ will be similar.

Now in these similar triangles corresponding sides will be in the same ratio.

$\frac{SP}{QT}=\frac{SO}{OQ}$

Since SO = 3OQ

Therefore, $\frac{SP}{QT}=3$

In the ΔPTQ and ΔUTR,

Segments SU and PQ are parallel and QR is the transverse,

Therefore, ∠URT ≅ ∠PQT [Alternate angles]

Segments SU and PQ are parallel and PU is the transverse,

Therefore, ∠TUR ≅ ∠TPQ [Alternate angles]

∠RTU ≅ ∠PTQ [Vertically opposite angles]

Since all three angles are equal therefore, ΔPTQ and ΔUTR will be similar.

By the property of similar triangles, corresponding sides of the triangles will be in the same ratio.

$\frac{RU}{PQ}=\frac{TR}{QT}$

     = $\frac{RQ-QT}{QT}$

     = $\frac{RQ}{QT}-1$

     = $\frac{SP}{QT}-1$ [Since RQ = SP, Sides of a parallelogram]

     = $\frac{3QT}{QT}-1$

     = 3 - 1

     = 2

$\frac{PQ}{RU}=\frac{1}{2}$

The answer matches with Option B.

Learn more about similar triangles from https://brainly.in/question/5032868