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Question
Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.
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Solution
We have to prove that the sum of four sides of quadrilateral is greater than sum of diagonal.
Since the sum of two sides of triangle is greater than third side.
In ΔPQR we have
PQ + QR > PR ..........(1)
In ΔRSPwe have
RS + SP >PR ..........(2)
In ΔPQS we have
PQ + SP > QS ........(3)
In ΔQRSwe have
QR + RS > QS .........(4)
Adding (1) & (2) & (3) and (4) we get
2(PQ + QR + RS + SQ ) >2 (PR + QS)
Hence (PQ + QR + RS + SQ > PR + QS)Proved.
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