In the given question we are asked to prove that AO/BO = CO/DO by using basic proportionality theorem, if ABCD is a trapezium in which AB∣∣DC and its diagonals intersect each other at the point O.

ML Aggarwal Avichal Publication class 10, Similarity, chapter 13.2, question no 8

Given parameters

ABCD is a trapezium where AB || DC and diagonals AC and BD intersect at O.

To proveAO/BO=CO/DO

ConstructionDraw a line EF passing through O and also parallel to AB

Now, AB ll CD

By construction EF ll AB

∴ EF ll CD

Consider the ΔADC,

Where EO ll AB

According to basic proportionality theorem

AE/ED=AO/OC ……………………(1)

Now consider Δ ABD

where EO ll AB

According to basic proportionality theorem

AE/ED=BO/OD …………………..(2)

From equation (1) and (2) we have

AO/OC=BO/OD

⇒ AO/BO=OC/OD