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. 

Answers (1)
S seema garhwal

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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