Question

If = {| ∈ , < 5} and = {| ∈ , 2 < 25}. Show that =

Answer 1

Answer:

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Step-by-step explanation:

In a quadrilateral ABCD, the diagonals AD and BC meet at O. If it is given that OA = OC

and OB = OD, prove that BC = AD and that ∠ACB = ∠CAD.

आकाशवाणी (अंग्रेज़ी: All India Radio) भारत के सूचना एवं प्रसारण मंत्रालय के अधीन संचालित सार्वजनिक क्षेत्र की रेडियो प्रसारण सेवा है।

⇒ 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.