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प्रश्न
The diagonals of a quadrilateral ABCD intersect each other at the point O such that `("AO")/("BO") = ("CO")/("DO")`. Show that ABCD is a trapezium.
उत्तर
Let us consider the following figure for the given question.
Draw a line OE || AB
In ΔABD, OE || AB
By using basic proportionality theorem, we obtain
`("AE")/("ED") = ("BO")/("OD")` ...(1)
However, it is given that
`("AO")/("OC") = ("OB")/("OD")` ...(2)
From equation (1) and (2) we obtain
`("AE")/("ED") = ("AO")/("OC")`
⇒ EO || DC ...[By the converse of basic proportionality theorem]
⇒ AB || OE || DC
⇒ AB || CD
∴ ABCD is a trapezium.
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