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Question

In the quadrilateral ABCD, the diagonals AC and BD are equal and perpendicular to each other. What type of a quadrilateral is ABCD?

A
A square
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B
A parallelogram
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C
A rectangle
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D
A trapezium
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Solution

The correct option is A A square⇒ In given figure ABCD is a square having all sides are equal and opposite sides parallel to each other. AC and BD are diagonals.⇒ In △ABC and △BAD,⇒ AB = AB [Common line]⇒ BC = AD [Sides of square are equal.]⇒ ∠ABC = ∠BAD [All four angles of square is 90∘]⇒ △ABC ≅ △BAD [By SAS property]⇒ In a △ OAD and △OCB,⇒ AD = CB [sides of a square]⇒ ∠OAD = ∠OCB [Alternate angle]⇒ ∠ODA = ∠OBC [Alternate angle]⇒ △OAD ≅ △OCB [By ASA Property]⇒ So, OA = OC ----- ( 1 )⇒ Similarly, OB = OD ------ ( 2 )From ( 1 ) and ( 2 ) we get that AC and BD bisect each other.⇒ Now, in △OBA and △ODA,⇒ OB = OD [From ( 2 )]⇒ BA = DA⇒ OA = OA [Common line]⇒ ∠AOB + ∠AOD ----- ( 3 ) [By CPCT]⇒ ∠AOB + ∠AOD = 180∘ [Linear pair]⇒ 2∠AOB = 180∘∴ ∠AOB = ∠AOD = 90∘∴ We have proved that diagonals of square are equal and perpendicular to each other.

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