Question

the diagonal of a quadrilateral abcd intersect each other at the point of such that ao/bo= co/do.show that abcd is trapezium​

Answer 1

Answer:

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Step-by-step explanation:

Given:

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

i.e., COAO=DOBO

To Prove: ABCD is a trapezium

Construction:

Draw OE∥DC such that E lies on BC.

Proof:

In △BDC,

By Basic Proportionality Theorem,

ODBO=ECBE............(1)

But, COAO=DOBO (Given) .........(2)

∴ From (1) and (2)

COAO=ECBE

Hence, By Converse of Basic Proportionality Theorem,

OE∥AB

Now Since, AB∥OE∥DC

∴ AB∥DC

Hence, ABCD is a trapezium.

Answer 2

Answer:

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.

solution