Question

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that `(O A)/(O C)=(O B)/(O D)`

Answer 1

GIVEN A trapezium. ABCD in which `AB||DC`. The diagonals AC and BD intersect at O.
TO PROOVE `(OA)/(OC)=(OB)/(OD)`
PROOF In `Delta OAB and Delta OCD`, we have

`angle OAB= angle OCD` [ alternate angels, since `AB||DC`]
and `angle OAB= angle ODC` [ alternative angles since `AB||DC`]
`:. Delta OAB ~ Delta OCD` [ by AA- similarity]
And so, `(OA)/(OC)=(OB)/(OD)`