Q. 14.4( 46 Votes )

Prove that the su

Answer :

Need to prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals



ABCD is a rhombus in which diagonals AC and BD intersect at point O.


We need to prove AB2 + BC2 + CD2 + DA2 = AC2 + DB2


In Δ AOB; AB2 = AO2 + BO2


In Δ BOC; BC2 = CO2 + BO2


In Δ COD; CD2 = DO2 + CO2


In Δ AOD; AD2 = DO2 + AO2


Adding the above 4 equations we get


AB2 + BC2 + CD2 + DA2 = AO2 + BO2 + CO2 + BO2 + DO2 + CO2 + DO2 + AO2


= 2(AO2 + BO2 + CO2 + DO2)


Since, AO2 = CO2 and BO2 = DO2


= 2(2 AO2 + 2 BO2)


= 4(AO2 + BO2) ……eq(1)


Now, let us take the sum of squares of diagonals


AC2 + DB2 = (AO + CO)2 + (DO+ BO)2


= (2AO)2 + (2DO)2


= 4 AO2 + 4 BO2 ……eq(2)


From eq(1) and eq(2) we get


AB2 + BC2 + CD2 + DA2 = AC2 + DB2


Hence, proved


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