Question

show that if the diagonal of a quadrilateral are equal and bisect each other at Right angle . then it is square​

Answer 1

Given: ABCD is a quadrilateral in which diagonals bisect each other at 90°

OB=OD

AO=OC

Answer 2

$\huge\star{\blue{\underline{\underline{\tt{Explanation:}}}}}$

______________________________

Given that,

• Let ABCD be a quadrilateral

• It's iagonals AC and BD bisect each other at right angle at O.

To prove that

• The Quadrilateral ABCD is a square.

Proof,

In ΔAOB and ΔCOD,

$\leadsto$AO = CO (Diagonals bisect each other)

$\leadsto$∠AOB = ∠COD (Vertically opposite)

$\leadsto$OB = OD (Diagonals bisect each other)

,$\leadsto$ ΔAOB ≅ ΔCOD [SAS congruency]

Thus,

$\leadsto$AB = CD [CPCT] — (i)

also,

∠OAB = ∠OCD (Alternate interior angles)

⇒ AB || CD

Now,

$\leadsto$In ΔAOD and ΔCOD,

$\leadsto$AO = CO (Diagonals bisect each other)

$\leadsto$∠AOD = ∠COD (Vertically opposite)

$\leadsto$OD = OD (Common)

$\leadsto$ 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.

$\large\star{\tt{\red{\underline{Hence\:Proved!}}}}$