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Question

In the figure given below a quadrilateral ABCD is draw to circumscribe a circle prove that :AB+CD=AD+BC
1508265_c1217c7e1bc2427595e83c6e1c242ceb.png

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Solution

Given:- Let ABCD be the quadrilateral circumscribing the circle with centre O. The quadrilateral touches the circle at point P,Q,R and S.
To prove:- AB+CD=AD+BC
Proof:-
AP=AS.....(1)(Length of tangents drawn from external point are equal)
Similarly,
BP=BQ.....(2)
CR=CQ.....(3)
DR=DS.....(4)
Adding equation (1),(2),(3)&(4), we have
AP+BP+CR+DR=AS+BQ+CQ+DS
(AP+PB)+(CR+RD)=(AS+SD)+(BQ+QC)
AB+CD=AD+BC
Hence proved.

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