Question

10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that
AO CO
Show that ABCD is a trapezium.
BO DO​

Answer 1

Step-by-step explanation:

Given:

The diagonals of a quadrilateral ABCD intersect each other at the point O such that

BO

AO

=

DO

CO

i.e.,

CO

AO

=

DO

BO

To Prove: ABCD is a trapezium

Construction:

Draw OE∥DC such that E lies on BC.

Proof:

In △BDC,

By Basic Proportionality Theorem,

OD

BO

=

EC

BE

............(1)

But,

CO

AO

=

DO

BO

(Given) .........(2)

∴ From (1) and (2)

CO

AO

=

EC

BE

Hence, By Converse of Basic Proportionality Theorem,

OE∥AB

Now Since, AB∥OE∥DC

∴ AB∥DC

Hence, ABCD is a trapezium.