Q

# ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point, O. Show, that AO / BO = CO / DO

Q9  ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show
that $\frac{AO}{BO} = \frac{CO}{DO}$

Views

Draw a line EF passing through point O such that $EO||CD\, \, and\, \, FO||CD$

To prove :

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

In $\triangle ADC$, we have $CD||EO$

So, by using basic proportionality theorem,

$\frac{AE}{ED}=\frac{AO}{OC}........................................1$

In $\triangle ABD$, we have $AB||EO$

So, by using basic proportionality theorem,

$\frac{DE}{EA}=\frac{OD}{BO}........................................2$

Using equation 1 and 2, we get

$\frac{AO}{OC}=\frac{BO}{OD}$

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

Hence proved

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