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Question

ABCD is a rectangle. Points M and N are on BD such that AM ⊥ BD and CN ⊥ BD. Prove that BM2 + BN2 = DM2 + DN2.

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Solution

Given: A rectangle ABCD where AM BD and CN BD.

To prove: BM2 + BN2 = DM2 + DN2

Proof:

Apply Pythagoras Theorem in ΔAMB and ΔCND,

AB2 = AM2 + MB2

CD2 = CN2 + ND2

Since AB = CD, AM2 + MB2 = CN2 + ND2

⇒ AM2 − CN2 = ND2 − MB2 … (i)

Again apply Pythagoras Theorem in ΔAMD and ΔCNB,

AD2 = AM2 + MD2

CB2 = CN2 + NB2

Since AD = BC, AM2 + MD2 = CN2 + NB2

⇒ AM2 − CN2 = NB2 − MD2 … (ii)

Equating (i) and (ii),

ND2 − MB2 = NB2 − MD2

I.e., BM2 + BN2 = DM2 + DN2

This proves the given relation.


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