If the diagonals of a quadrilateral are equal and bisect each other at right angles, then prove that it is a square.
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Question

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

KnowledgeBoat · Accepted Answer

Since, diagonals bisect each other at 90° ∴ ∠AOB = ∠COD = ∠BOC = AOD = 90°. From figure, Considering △OAB and △ODC we have, ⇒ OA = OC (As diagonals bisect each other) ⇒ OB = OD (As diagonals bisect each other) ⇒ ∠AOB = ∠COD (Vertically opposite angles are equal) Hence, △OAB ≅ △ODC by SAS axiom. AB = CD (By C.P.C.T.) .........(i) ∴ ∠OAB = ∠OCD (By C.P.C.T.) The above angles are alternate angles. Hence, we can say that AB || CD. In △AOB, OA = OB (As both diagonals are equal and bisect each other) so, ⇒ ∠OBA = ∠OAB = x (let) (Angles opposite to equal sides are equal in isosceles triangle) ⇒ ∠OBA + ∠OAB + ∠AOB = 180° ⇒ x + x + 90° = 180° ⇒ 2x = 180° - 90° ⇒ 2x = 90° ⇒ x = ⇒ x = 45° Considering △OAD and △OBC we have, ⇒ OA = OC (As diagonals bisect each other) ⇒ OB = OD (As diagonals bisect each other) ⇒ ∠AOD = ∠COB (Vertically opposite angles are equal) Hence, △OAD ≅ △OBC by SAS axiom. AD = BC (By C.P.C.T.) .........(ii) ∠OAD = ∠OCB (By C.P.C.T.) Hence, we can say that AD || BC. Considering △AOB and △AOD we have, ⇒ AO = AO (Common sides) ⇒ OB = OD (As diagonals bisect each other) ⇒ ∠AOD = ∠AOB (Both equal to 90°) Hence, △AOB ≅ △AOD by SAS axiom. AB = AD (By C.P.C.T.) .........(iii) From (i), (ii) and (iii) we get, AB = BC = CD = AD. In △AOD, OA = OD (As both diagonals are equal and bisect each other) so, ⇒ ∠OAD = ∠ODA = a (let) (Angles opposite to equal sides are equal in isosceles triangle) ⇒ ∠OAD + ∠ODA + ∠AOD = 180° ⇒ a + a + 90° = 180° ⇒ 2a = 180° - 90° ⇒ 2a = 90° ⇒ a = ⇒ a = 45°. ∠A = ∠OAB + OAD = 45° + 45° = 90°. Thus, AB ⊥ AD. Since, AD || BC so, AB ⊥ BC. Since, AB || CD and AB ⊥ AD, ∴ CD ⊥ AD. Since, alternate sides are perpendicular and all sides are equal. Hence, proved that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square

Mathematics

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

Rectilinear Figures

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