Hello sir i want to know the best solution of the question from exercise 8.1of math of Quadrilaterals chapter of class 9th give me the best and easy for solving this question how i solve it of question no. 5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
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Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. Q.5
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Given that,
Let ABCD be a quadrilateral and its diagonals 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.
Hence Proved.