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# Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.

Q2   Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O.
If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.

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Given: Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O.

AB = 2 CD          ( Given )

In $\triangle AOB\, and\, \triangle COD,$

$\angle COD=\angle AOB$                (vertically opposite angles )

$\angle OCD=\angle OAB$                 (Alternate angles)

$\angle ODC=\angle OBA$                 (Alternate angles)

$\therefore \triangle AOB\, \sim \, \triangle COD$               (AAA similarity)

$\frac{ar(\triangle AOB)}{ar(\triangle COD)}=\frac{AB^2}{CD^2}$

$\frac{ar(\triangle AOB)}{ar(\triangle COD)}=\frac{(2CD)^2}{CD^2}$

$\frac{ar(\triangle AOB)}{ar(\triangle COD)}=\frac{4.CD^2}{CD^2}$

$\Rightarrow \frac{ar(\triangle AOB)}{ar(\triangle COD)}=\frac{4}{1}$

$\Rightarrow ar(\triangle AOB)=ar(\triangle COD)=4:1$

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