Question

In a trapezium ABCD, AB || DC and its diagonals intersect at O. Prove that `(OA)/(OC) = (OB)/(OD)`.

Answer 1

Given: In a trapezium ABCD, AB || DC.

Also, AC and BD intersect each other at O.

Construction: Draw XY || AB and CD through ‘Ο’.

To prove: `(OA)/(OC) = (OB)/(OD)`

Proof: In ΔABC, OY || AB

`(BY)/(YC) = "AO"/"OC"`   ...(1) (By BPT)

In ΔBCD, OY || CD

`(BY)/(YC) = (OB)/(OD)`   ...(2) (By BPT)

From equation (1) and (2)

`(OA)/(OC) = (OB)/(OD)`

Hence proved.