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 the question with explanation please...........

Answer 1

Answer:

Do the answer by yourself

Answer 2

Answer:

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

Step-by-step explanation:

In the parallelogram PQRS, sides SP and RQ are the parallel sides and SQ is the transverse.

Therefore, ∠SQR ≅ ∠PSQ  [Alternate angles]

Sides SP and RQ are parallel and PU is the transverse.

∠SOP ≅ ∠TOQ [Vertically opposite angles]

Therefore, ΔSOP ~ ΔTOQ

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

$\frac{SO}{OQ}=\frac{SP}{QT}=\frac{3}{1}$

SP = 3×QT ---------(1)

Now in ΔPTQ and ΔUTR,

Side PQ ║ SR and TQ is a transverse, so ∠URT ≅ ∠PQT [Alternate angles]

Side SU ║ PQ and PU is a transverse, so ∠RUT ≅ ∠QPT [Alternate angles]

∠RTU ≅ ∠PTQ [Vertically opposite angles]

Therefore, ΔPTQ ≅ ΔUTR will be similar.

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

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

     = $\frac{QT}{RQ-QT}$

$\frac{RU}{PQ}=\frac{RQ-QT}{QT}$

     = $\frac{RQ}{QT}-1$

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

     = $\frac{3QT}{QT}-1$

     = (3 - 1)

     = 2

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

Therefore, Option B. ($\frac{1}{2}$) will be the answer.

Learn more about properties of similar triangles from https://brainly.in/question/5004598