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

Answers (1)

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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