Mathematics
In a rectangle ABCD, prove that :
AC2 + BD2 = AB2 + BC2 + CD2 + DA2.
Pythagoras Theorem
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Answer
In rectangle the interior angles equal to 90° and opposite sides are equal.
∴ ∠A = ∠B = ∠C = ∠D = 90°, AB = DC and BC = AD.
By formula,
By pythagoras theorem,
⇒ (Hypotenuse)2 = (Perpendicular)2 + (Base)2
In right-angled △ ACD,
⇒ AC2 = AD2 + CD2 ……..(1)
In right-angled △ BCD,
⇒ BD2 = BC2 + CD2
⇒ BD2 = BC2 + AB2 (As CD = AB) ………(2)
Adding equation (1) and (2), we get :
⇒ AC2 + BD2 = AD2 + CD2 + BC2 + AB2.
Hence, proved that AC2 + BD2 = AB2 + BC2 + CD2 + DA2.
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