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Question
In a rectangle ABCD,
prove that: AC2 + BD2 = AB2 + BC2 + CD2 + DA2.
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Solution
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Since, ABCD is a rectangle angles A, B, C and D are rt. angles.
First, we consider the ΔACD, and applying Pythagoras theorem we get,
AC2 = DA2 + CD2 ....(i)
Similarly, we get from rt. angle triangle BDC we get,
BD2 = BC2 + CD2
= BC2 + AB2 ....[ In a rectangle, opposite sides are equal, ∴ CD = AB ] ...(ii)
Adding (i) and (ii)
AC2 + BD2 = AB2 + BC2 + CD2 + DA2
Hence proved.
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