Q10 of 53 Page 153

The diagonals of a convex â ABCD intersect at right angles. Prove that AB2 + CD2 = AD2 + BC2.

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Using Pythagoras theorem :


AO2 + OB2 = AB2 …Equation(i)


DO2 + OC2 = CD2 …Equation(ii)


AO2 + OD2 = AD2 …Equation(iii)


BO2 + OC2 = BC2 …Equation(iv)


Adding Equation(i) and Equation(ii)


AB2 + CD2 = AO2 + OB2 + DO2 + OC2 …Equation(v)


Adding Equation(iii) and Equation(iv)


AD2 + BC2 = AO2 + OB2 + DO2 + OC2 …Equation(vi)


The RHS of equation (v) and (vi) are similar


So we can say that


AB2 + CD2 = AD2 + BC2


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