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