Question

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prove that the diagonals of rhombus bisect each other at right angles


$\huge{\green{\underline{\underline{\tt{❌Don't Span❌}}}}}$​

Answer 1

$\huge\underline\mathcal\orange{ᴀɴꜱᴡᴇʀ}$

Let ABCD is a rhombus.

⇒ AB=BC=CD=DA [ Adjacent sides are eqaul in rhombus ]

In △AOD and △COD

⇒ OA=OC [ Diagonals of rhombus bisect each other ]

⇒ OD=OD [ Common side ]

⇒ AD=CD

∴ △AOD≅△COD [ By SSS congruence rule ]

⇒ ∠AOD=∠COD [ CPCT ]

⇒ ∠AOD+∠COD=180

o

[ Linear pair ]

⇒ 2∠AOD=180

o

.

∴ ∠AOD=90

o

.

Hence, the diagonals of a rhombus bisect each other at right angle.

Answer 2

$\huge \color{lime}ANSWER :$

Let ABCD is a rhombus.

⇒ AB=BC=CD=DA [ Adjacent sides are eqaul in rhombus ]

In △AOD and △COD

⇒ OA=OC [ Diagonals of rhombus bisect each other ]

⇒ OD=OD [ Common side ]

⇒ AD=CD

∴ △AOD≅△COD [ By SSS congruence rule ]

⇒ ∠AOD=∠COD [ CPCT ]

⇒ ∠AOD+∠COD=180

o

[ Linear pair ]

⇒ 2∠AOD=180

o

.

∴ ∠AOD=90

o

.

Hence, the diagonals of a rhombus bisect each other at right angle.