Question

The diagonals of trapezium PQRS intersect each other at O. Prove that: `(PO)/(OR) = (QO)/(OS)`

Answer 1

Since PQ || SR, and the diagonals PR and QS intersect at O:

Consider triangles △POQ and △ROS.

We have:

• ∠POQ = ∠ROS because they are vertically opposite angles.
• ∠PQO = ∠RSO because PQ || SR, and QO and SO lie on the same line QS.

Therefore, △POQ ∼ △ROS by AA similarity.

So, the corresponding sides are proportional:

`(PO)/(OR) = (QO)/(OS) = (PQ)/(RS)`

Thus, `(PO)/(OR) = (QO)/(OS)`

Hence proved.