Answer :
Given: AC ⊥ BD
To prove: AB2 + CD2 = BC2 + DA2
Proof:
By applying Pythagoras theorem in ΔAOB, we get,
AB2 = AO2 + BO2 ……………… (1)
By applying Pythagoras theorem in ΔBOC, we get,
BC2 = BO2 + CO2 ……………… (2)
By applying Pythagoras theorem in ΔCOD, we get,
CD2 = CO2 + DO2 ……………… (3)
By applying Pythagoras theorem in ΔDOA, we get,
DA2 = AO2 + DO2 ……………… (4)
By adding eqn. (1) and (3), we get,
AB2 + CD2 = AO2 + BO2 + CO2 + DO2
⇒ AB2 + CD2 = (BO2 + CO2) + (AO2 + DO2)
Substituting from eqn. (2) and (4) gives,
AB2 + CD2 = BC2 + DA2
Hence proved.
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