Question

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.

Answer 1

Given: The diagonals of a quadrilateral ABCD intersect each other at the point O, such that \(\frac{AO}{BO}=\frac{CO}{DO}\)

To Show: ABCD is a tapezium

Solution: Let us consider the following figure for the given question

Draw a line OE || AB

In ∆ABD, OE || AB

By using the basic proportionality theorem, we obtain  
\(\frac{AE}{ED}=\frac{BO}{DO}\) .....(i)

However, it is given that  
\(\frac{AO}{OC}=\frac{OB}{OD}\) ......(ii)

From Equation (i) and (ii) we obtain,
\(\frac{AE}{ED}=\frac{AO}{OC}\)
⇒ EO || DC [By the converse of basic proportionality theorem]  
⇒ AB || OE || DC  
⇒ AB || CD  
∴ ABCD is a trapezium.