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 prove
AO/BO=CO/DO
Construction
Draw 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