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Diagonals of a rectangle are equal and perpendicular. Is this statement True or False?  Give reason for your answer.

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Answer: False

Solution.
Given that diagonals of a rectangle are equal and perpendicular.
Rectangle: A rectangle an equiangular quadrilateral, and all of its angles are equal.
Hence diagonals of a rectangle are equal but not necessary perpendicular to each other.
Let us consider a rectangle ABCD

Consider
\triangle ACD and \triangle BCD
AC = BD (opposite sides of a rectangle are equal)
\angle C=\angle D                  (90^{\circ})
AB = CD
(opposite sides of a rectangle are equal)
\triangle ACD\cong \triangle BCD            (SAS congruency)
So, AD = BC
Hence diagonals are equal.
Also, \angle CAD=\angle DBC               …(i)
Similarly, we can prove that \triangle ACB and
\triangle BDA are congruent
Hence, \angle ACB=\angle ADB           …(ii)
Now, consider
\triangle AOC and \triangle BOD
\angle CAD=\angle DBC        From (i)
\angle ACB=\angle ADB                   From (ii)
\angle AOC=\angle BOD                  vertically opposite angles
\triangle AOC and \triangle BOD 
are also congruent.

But we cannot prove that \angle AOC=\angle BOD=90^{\circ}
Hence it is not necessary that diagonals will bisect each other at right angle, so they are not necessarily perpendicular to each other
Hence the given statement is False.

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