Sum no. 10 Exercise 6.2 class 10th



Sum No.10:-
The diagonals of  a quadrilateral ABCD intersect each other at a point O such that AO/BO=CO/DO.
show that ABCD is a trapezium.
Solution:-
Given:- ABCD is a quadrilateral, whose diagonals AC and BD intesect each other at O.
such that AO/BO=CO/DO i.e., AO/CO=BO/DO......(1)
To prove :- ABCD is a trapezium.
Construction:- from O,draw OE∥AB.

Proof:- In △ADB, we have,
             OE∥AB (construction)
∴ by BPT, we have 
AE/ED=BO/DO.....(2)
from (1) and (2),we have AO/CO=AE/ED
thus in △ADC,E and O are points dividing the sides AD and AC in the same ratio.
∴ by converse of BPT, we have
{OE∥CD but OE∥AB by construction}
hence OE∥CD∥AB ⇒ AB∥CD
∴ Quadrilatral ABCD is at Trapezium.

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