Question -
Answer -
Given, Quadrilateral ABCD where AC and BDintersects each other at O such that,
AO/BO = CO/DO.
We have to prove here, ABCD is a trapezium
From the point O, draw a line EO touching ADat E, in such a way that,
EO || DC || AB
In ΔDAB, EO || AB
Therefore, By using Basic ProportionalityTheorem
DE/EA = DO/OB ……………………(i)
Also, given,
AO/BO = CO/DO
⇒ AO/CO = BO/DO
⇒ CO/AO = DO/BO
⇒DO/OB = CO/AO …………………………..(ii)
From equation (i) and (ii),we get
DE/EA = CO/AO
Therefore, By using converse of BasicProportionality Theorem,
EO || DC also EO || AB
⇒ AB || DC.
Hence, quadrilateral ABCD is a trapezium withAB || CD.