Question

ABCD is a trapezium in which AB is parallel to DC. If the diagonals intersect at O, then which one of the following is correct?
A. $\dfrac{{OA}}{{OC}} = \dfrac{{OB}}{{OD}}$
B. $\dfrac{{AD}}{{BC}} = \dfrac{{AB}}{{DC}}$
C. $\dfrac{{OB}}{{OD}} = \dfrac{{BC}}{{CD}}$
D. $\dfrac{{OA}}{{OC}} = \dfrac{{DA}}{{DC}}$

Answer 1

Hint: In this question use the properties of trapezium to find the correct option and they are: The bases of trapezoid are parallel, one of the pairs of opposite sides are parallel, each lower base angle is supplementary to upper base angle on the same side.

Complete step by step answer:

According to question it is given that,
In a trapezium ABCD, the diagonal intersects each other at O. In triangle OAB and OCD,
$\angle AOB = \angle COD$(opposite angles)
$\angle ABO = \angle ODC$(alternate angles on parallel lines)
$\angle BAO = \angle OCD$(alternate angles on parallel lines)
Therefore, triangle $OAB$and $OCD$are similar.
Hence $\dfrac{{OA}}{{OB}} = \dfrac{{OC}}{{OD}}$
OR $\dfrac{{OA}}{{OC}} = \dfrac{{OB}}{{OD}}$

Note:
 It is always advisable to remember such properties while involving these geometric questions as it helps save a lot of time. Eventually it will be difficult to learn all the properties of every shape but with practice things get easier.