Mathematics
Assertion (A): The diagonal of a quadrilateral bisect each other at right angle.
Reason (R): The quadrilateral is square.
A is true, but R is false.
A is false, but R is true.
Both A and R are true, and R is the correct reason for A.
Both A and R are true, and R is the incorrect reason for A.
Rectilinear Figures
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Answer
Let quadrilateral be ABCD.
Since, diagonals bisect each other at 90°.
∴ Assertion (A) is true.
∴ ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°.
From figure,
Considering △OAB and △OCD we have,
⇒ OA = OC (As diagonals bisect each other)
⇒ OB = OD (As diagonals bisect each other)
⇒ ∠AOB = ∠COD (Both equal to 90°)
Hence, △OAB ≅ △OCD by SAS axiom.
AB = CD (By C.P.C.T.) …………………….(1)
∴ ∠OAB = ∠OCD (By C.P.C.T.)
The above angles are alternate angles.
Hence, we can say that AB || CD.
Considering △OAD and △OCB we have,
⇒ OA = OC (As diagonals bisect each other)
⇒ OB = OD (As diagonals bisect each other)
⇒ ∠AOD = ∠COB (Both equal to 90°)
Hence, △OAD ≅ △OCB by SAS axiom.
AD = BC (By C.P.C.T.) …………………(2)
∠OAD = ∠OCB (By C.P.C.T.)
The above angles are alternate angles.
Hence, we can say that AD || BC.
Considering △AOB and △AOD we have,
⇒ AO = AO (Common side)
⇒ 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.) …………………(3)
From (i), (ii) and (iii) we get,
AB = BC = CD = AD.
Since, all the sides are equal and diagonals bisect each other.
Thus, we can say that the quadrilateral is rhombus.
∴ A is true, but R is false.
Hence, option 1 is the correct option.
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