Question
The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.
✓
Answer
To prove: $\text{ABCD}$ is a square,
that is, to prove that sides of the quadrilateral are equal
and each angle of the quadrilateral is $90^{\circ}$,
$\text{ABCD}$ is a rectangle,
$\Rightarrow \angle A =\angle B =\angle C =\angle D =90^{\circ}$ and diagonals bisect each other.
that is, $MD = BM \dots....(i)$
Consider $\triangle AMD$ and $\triangle AMB$,
$MD = BM \dots....($ from $(i))$
$\angle AMD =\angle AMB =90^{\circ} \dots.....$(given$)$
$AM = AM \dots......($common side $)$
$\Rightarrow A D=A B \dots....( \text{c.p.c.t.c.} )$
Since $\text{ABCD}$ is a rectangle, $A D=B C$ and $A B=C D$
Thus, $A B=B C=C D=A D$ and $\angle A=\angle B=\angle C=\angle D=90^{\circ}$
$\Rightarrow \text{ABCD}$ is a square.
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