1.

Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.​

Answer»

\large\sf\underline\red{Given:-}

The diagonals of a quadrilateral are equal and BISECT each other at right angles, then it is a square.

\large\sf\underline\green{Explanation:-}

Let ABCD be a quadrilateral and its diagonals AC and BD bisect each other at right ANGLE at O.

\large\sf\underline\pink{To\:prove:-}

The Quadrilateral ABCD is a square.

\large\sf\underline\purple{Proof:-}

In ΔAOB and ΔCOD,

AO = CO (Diagonals bisect each other)

∠AOB = ∠COD (Vertically opposite)

OB = OD (Diagonals bisect each other)

ΔAOB ≅ ΔCOD [SAS congruency]

__________________

Thus,

AB = CD [CPCT] — (i)

also,

∠OAB = ∠OCD (Alternate interior angles)

⇒ AB || CD

\large\sf\underline\orange{Now:-}

In ΔAOD and ΔCOD,

AO = CO (Diagonals bisect each other)

∠AOD = ∠COD (Vertically opposite)

OD = OD (Common)

ΔAOD ≅ ΔCOD [SAS congruency]

_________________

Thus,

AD = CD [CPCT] — (ii)

Also,

AD = BC and AD = CD

⇒ AD = BC = CD = AB — (ii)

also, ∠ADC = ∠BCD [CPCT]

and ∠ADC + ∠BCD = 180° (co-interior angles)

⇒ 2∠ADC = 180°

⇒ ∠ADC = 90° — (iii)

One of the interior angles is right angle.

Thus, from (i), (ii) and (iii) given quadrilateral ABCD is a square.

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