Mathematics
In a quadrilateral ABCD, ∠A + ∠D = 90°, prove that : AC2 + BD2 = AD2 + BC2.
Pythagoras Theorem
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Answer
Produce AB and DC such that they meet at point E.
By formula,
By pythagoras theorem,
⇒ (Hypotenuse)2 = (Perpendicular)2 + Base2
In △ AED,
⇒ ∠A + ∠D + ∠E = 180°
⇒ 90° + ∠E = 180°
⇒ ∠E = 180° - 90° = 90°.
By pythagoras theorem,
⇒ AD2 = AE2 + DE2 ………(1)
In △ BEC,
By pythagoras theorem,
⇒ BC2 = BE2 + CE2 ………(2)
In △ AEC,
By pythagoras theorem,
⇒ AC2 = AE2 + CE2 ………(3)
In △ BED,
By pythagoras theorem,
⇒ BD2 = BE2 + DE2 ………(4)
Adding equations (1) and (2), we get :
⇒ AD2 + BC2 = AE2 + DE2 + BE2 + CE2
⇒ AD2 + BC2 = (AE2 + CE2) + (BE2 + DE2)
⇒ AD2 + BC2 = AC2 + BD2.
Hence, proved that AC2 + BD2 = AD2 + BC2.
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