Diagonals of a trapezium divide each other proportionally , Mathematics

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Diagonals of a trapezium divide each other proportionally:

Given : In trapezium ABCD , AB// DC
R.T.P :AO/OC = BO/OD

Construction: Draw the line PQ; parallel to AB or CD through O
Proof : In ΔACD, OP// CD

AP/PD = AO/OC................... (1)

(Basic proportionality theorem)

In ΔABD, OP// AB

AP/PD = BO/OD.................(2)

(Basic proportionality theorem)

From (1) and (2) we have

AO/OC = BO/OD

i.e, The diagonals AC, BD divide each other proportionally.

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