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

Q10   The diagonals of a quadrilateral ABCD intersect each other at the point O such that
$\frac{AO}{BO} = \frac{CO}{DO}$  Show that ABCD is a trapezium.

Views

Draw a line EF passing through point O such that $EO||AB$

Given  :

$\frac{AO}{BO} = \frac{CO}{DO}$

In $\triangle ABD$, we have $AB||EO$

So, by using basic proportionality theorem,

$\frac{AE}{ED}=\frac{BO}{DO}........................................1$

However, its is given that

$\frac{AO}{CO} = \frac{BO}{DO}..............................2$

Using equation 1 and 2 , we get

$\frac{AE}{ED}=\frac{AO}{CO}$

$\Rightarrow EO||CD$         (By basic proportionality theorem)

$\Rightarrow AB||EO||CD$

$\Rightarrow AB||CD$

Therefore, ABCD is a trapezium.

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