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  • Show that the diagonals of a square are equal and bisect each other at right angles.

Q: 4 Show that the diagonals of a square are equal and bisect each other at right angles.
 

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Given : ABCD is a square i.e. AB=BC=CD=DA.

To prove : the diagonals of a square are equal and bisect each other at right angles i.e. AC=BD,AO=CO,BO=DO and \angle COD=90 \degree

Proof : In \triangleBAD and \triangleABC,   

   

  \angle BAD = \angle ABC        (Each 90 \degree)

  AD=BC              (Given )

  AB=AB               (common)

  \triangleBAD \cong \triangleABC   (By SAS)

  BD=AC           (CPCT)

In \triangleAOB and \triangleCOD, 

 \angleOAB=\angleOCD        (Alternate angles)

 AB=CD               (Given )

 \angleOBA=\angleODC       (Alternate angles)

  \triangleAOB \cong \triangleCOD   (By AAS)

  AO=OC ,BO=OD           (CPCT)

In \triangleAOB and \triangleAOD, 

  OB=OD        (proved above)

  AB=AD               (Given )

   OA=OA          (COMMON)

  \triangleAOB \cong \triangleAOD   (By SSS)

  \angleAOB=\angleAOD           (CPCT)

  \angleAOB+\angleAOD =180 \degree

  2. \angleAOB = 180 \degree

  \angleAOB =90 \degree 

Hence,  the diagonals of a square are equal and bisect each other at right angles.

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seema garhwal

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