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

Name: Diagonals of a rectangle are equal and perpendicular. Is this statement True or False? Give reason for your answer.
Uploaded: Thu, 2021-01-07 14:06
Description: Diagonals of a rectangle are equal and perpendicular. Is this statement True or False? Give reason for your answer.

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

Answers (1)

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.