show that if the diagonals of a quadrilateral are equal and bisect each other at right angles,then it is a square.
Answers
Step-by-step explanation:
Explanation:
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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,
⇝ 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
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.
HenceProved!