Question

30. In a quadrilateral ABCD, given that angle A+ angle D= 90°. Prove that AC square + BD square = AD square + BC square ?​

Answer 1

Answer:

Given: In quadrilateral ABCD,

∠A+∠D=90∘

AC and BD are joined

To prove: AC² + BD² = AD² + BC²

Construction: Produce AB and DC to meet at P.

Proof: In △APD,

∠A+∠D=90∘(given)

∴∠P=90∘ (∵∠A+∠P+∠D=180∘)

Now in right △ACP,∠APD=90∘

AC² = PA² + PC² ....(i)

(Pythagoras Theorem)

and in △BPD

BD² = PB² + PD²

Adding (i) and (ii)

AC² + BD² = PA² + PC² + PB² + PD²

= (PA² + PD²) + (PC² + PB²)

= AD² + BD²

Hence proved.

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